{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0001","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"b0\",\"c0\"],[\"b0\"],[\"c0\"],[]],\"compact_subset\":[\"a0\",\"b0\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, b0, c0], [a0, b0], [[a0, b0, [[a0, c0], [b0]]], [a0, c0, [[a0, b0], [c0]]], [b0, c0, [[a0, b0], [c0]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a0, b0, c0], [a0, b0], [[a0, b0, [[a0, c0], [b0]]], [a0, c0, [[a0, b0], [c0]]], [b0, c0, [[a0, b0], [c0]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123001,"parameters":{"points":["a0","b0","c0"],"topology":[["a0","b0","c0"],["a0","b0"],["a0","c0"],["a0"],["b0","c0"],["b0"],["c0"],[]],"compact_subset":["a0","b0"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a0\",\"b0\"],\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"b0\",\"c0\"],[\"b0\"],[\"c0\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0002","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"b1\",\"c1\"],[\"b1\"],[\"c1\"],[]],\"compact_subset\":[\"a1\",\"b1\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a1, b1, c1], [a1, b1], [[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a1, b1, c1], [a1, b1], [[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123002,"parameters":{"points":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["b1","c1"],["b1"],["c1"],[]],"compact_subset":["a1","b1"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a1\",\"b1\"],\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"b1\",\"c1\"],[\"b1\"],[\"c1\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0003","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"b2\",\"c2\"],[\"b2\"],[\"c2\"],[]],\"compact_subset\":[\"a2\",\"b2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a2, b2, c2], [a2, b2], [[a2, b2, [[a2, c2], [b2]]], [a2, c2, [[a2, b2], [c2]]], [b2, c2, [[a2, b2], [c2]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a2, b2, c2], [a2, b2], [[a2, b2, [[a2, c2], [b2]]], [a2, c2, [[a2, b2], [c2]]], [b2, c2, [[a2, b2], [c2]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123003,"parameters":{"points":["a2","b2","c2"],"topology":[["a2","b2","c2"],["a2","b2"],["a2","c2"],["a2"],["b2","c2"],["b2"],["c2"],[]],"compact_subset":["a2","b2"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a2\",\"b2\"],\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"b2\",\"c2\"],[\"b2\"],[\"c2\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0004","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"b3\",\"c3\"],[\"b3\"],[\"c3\"],[]],\"compact_subset\":[\"a3\",\"b3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a3, b3, c3], [a3, b3], [[a3, b3, [[a3, c3], [b3]]], [a3, c3, [[a3, b3], [c3]]], [b3, c3, [[a3, b3], [c3]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a3, b3, c3], [a3, b3], [[a3, b3, [[a3, c3], [b3]]], [a3, c3, [[a3, b3], [c3]]], [b3, c3, [[a3, b3], [c3]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123004,"parameters":{"points":["a3","b3","c3"],"topology":[["a3","b3","c3"],["a3","b3"],["a3","c3"],["a3"],["b3","c3"],["b3"],["c3"],[]],"compact_subset":["a3","b3"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a3\",\"b3\"],\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"b3\",\"c3\"],[\"b3\"],[\"c3\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0005","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"b4\",\"c4\"],[\"b4\"],[\"c4\"],[]],\"compact_subset\":[\"a4\",\"b4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a4, b4, c4], [a4, b4], [[a4, b4, [[a4, c4], [b4]]], [a4, c4, [[a4, b4], [c4]]], [b4, c4, [[a4, b4], [c4]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a4, b4, c4], [a4, b4], [[a4, b4, [[a4, c4], [b4]]], [a4, c4, [[a4, b4], [c4]]], [b4, c4, [[a4, b4], [c4]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123005,"parameters":{"points":["a4","b4","c4"],"topology":[["a4","b4","c4"],["a4","b4"],["a4","c4"],["a4"],["b4","c4"],["b4"],["c4"],[]],"compact_subset":["a4","b4"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a4\",\"b4\"],\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"b4\",\"c4\"],[\"b4\"],[\"c4\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0006","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"b5\",\"c5\"],[\"b5\"],[\"c5\"],[]],\"compact_subset\":[\"a5\",\"b5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a5, b5, c5], [a5, b5], [[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a5, b5, c5], [a5, b5], [[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123006,"parameters":{"points":["a5","b5","c5"],"topology":[["a5","b5","c5"],["a5","b5"],["a5","c5"],["a5"],["b5","c5"],["b5"],["c5"],[]],"compact_subset":["a5","b5"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a5\",\"b5\"],\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"b5\",\"c5\"],[\"b5\"],[\"c5\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0007","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"b6\",\"c6\"],[\"b6\"],[\"c6\"],[]],\"compact_subset\":[\"a6\",\"b6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a6, b6, c6], [a6, b6], [[a6, b6, [[a6, c6], [b6]]], [a6, c6, [[a6, b6], [c6]]], [b6, c6, [[a6, b6], [c6]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a6, b6, c6], [a6, b6], [[a6, b6, [[a6, c6], [b6]]], [a6, c6, [[a6, b6], [c6]]], [b6, c6, [[a6, b6], [c6]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123007,"parameters":{"points":["a6","b6","c6"],"topology":[["a6","b6","c6"],["a6","b6"],["a6","c6"],["a6"],["b6","c6"],["b6"],["c6"],[]],"compact_subset":["a6","b6"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a6\",\"b6\"],\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"b6\",\"c6\"],[\"b6\"],[\"c6\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0008","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"b7\",\"c7\"],[\"b7\"],[\"c7\"],[]],\"compact_subset\":[\"a7\",\"b7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a7, b7, c7], [a7, b7], [[a7, b7, [[a7, c7], [b7]]], [a7, c7, [[a7, b7], [c7]]], [b7, c7, [[a7, b7], [c7]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a7, b7, c7], [a7, b7], [[a7, b7, [[a7, c7], [b7]]], [a7, c7, [[a7, b7], [c7]]], [b7, c7, [[a7, b7], [c7]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123008,"parameters":{"points":["a7","b7","c7"],"topology":[["a7","b7","c7"],["a7","b7"],["a7","c7"],["a7"],["b7","c7"],["b7"],["c7"],[]],"compact_subset":["a7","b7"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a7\",\"b7\"],\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"b7\",\"c7\"],[\"b7\"],[\"c7\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0009","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"b8\",\"c8\"],[\"b8\"],[\"c8\"],[]],\"compact_subset\":[\"a8\",\"b8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a8, b8, c8], [a8, b8], [[a8, b8, [[a8, c8], [b8]]], [a8, c8, [[a8, b8], [c8]]], [b8, c8, [[a8, b8], [c8]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a8, b8, c8], [a8, b8], [[a8, b8, [[a8, c8], [b8]]], [a8, c8, [[a8, b8], [c8]]], [b8, c8, [[a8, b8], [c8]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123009,"parameters":{"points":["a8","b8","c8"],"topology":[["a8","b8","c8"],["a8","b8"],["a8","c8"],["a8"],["b8","c8"],["b8"],["c8"],[]],"compact_subset":["a8","b8"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a8\",\"b8\"],\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"b8\",\"c8\"],[\"b8\"],[\"c8\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0010","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"b9\",\"c9\"],[\"b9\"],[\"c9\"],[]],\"compact_subset\":[\"a9\",\"b9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a9, b9, c9], [a9, b9], [[a9, b9, [[a9, c9], [b9]]], [a9, c9, [[a9, b9], [c9]]], [b9, c9, [[a9, b9], [c9]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a9, b9, c9], [a9, b9], [[a9, b9, [[a9, c9], [b9]]], [a9, c9, [[a9, b9], [c9]]], [b9, c9, [[a9, b9], [c9]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123010,"parameters":{"points":["a9","b9","c9"],"topology":[["a9","b9","c9"],["a9","b9"],["a9","c9"],["a9"],["b9","c9"],["b9"],["c9"],[]],"compact_subset":["a9","b9"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a9\",\"b9\"],\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"b9\",\"c9\"],[\"b9\"],[\"c9\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0011","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"b10\",\"c10\"],[\"b10\"],[\"c10\"],[]],\"compact_subset\":[\"a10\",\"b10\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a10, b10, c10], [a10, b10], [[a10, b10, [[a10, c10], [b10]]], [a10, c10, [[a10, b10], [c10]]], [b10, c10, [[a10, b10], [c10]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a10, b10, c10], [a10, b10], [[a10, b10, [[a10, c10], [b10]]], [a10, c10, [[a10, b10], [c10]]], [b10, c10, [[a10, b10], [c10]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123011,"parameters":{"points":["a10","b10","c10"],"topology":[["a10","b10","c10"],["a10","b10"],["a10","c10"],["a10"],["b10","c10"],["b10"],["c10"],[]],"compact_subset":["a10","b10"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a10\",\"b10\"],\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"b10\",\"c10\"],[\"b10\"],[\"c10\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0012","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"b11\",\"c11\"],[\"b11\"],[\"c11\"],[]],\"compact_subset\":[\"a11\",\"b11\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a11, b11, c11], [a11, b11], [[a11, b11, [[a11, c11], [b11]]], [a11, c11, [[a11, b11], [c11]]], [b11, c11, [[a11, b11], [c11]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a11, b11, c11], [a11, b11], [[a11, b11, [[a11, c11], [b11]]], [a11, c11, [[a11, b11], [c11]]], [b11, c11, [[a11, b11], [c11]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123012,"parameters":{"points":["a11","b11","c11"],"topology":[["a11","b11","c11"],["a11","b11"],["a11","c11"],["a11"],["b11","c11"],["b11"],["c11"],[]],"compact_subset":["a11","b11"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a11\",\"b11\"],\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"b11\",\"c11\"],[\"b11\"],[\"c11\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0013","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"b12\",\"c12\"],[\"b12\"],[\"c12\"],[]],\"compact_subset\":[\"a12\",\"b12\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a12, b12, c12], [a12, b12], [[a12, b12, [[a12, c12], [b12]]], [a12, c12, [[a12, b12], [c12]]], [b12, c12, [[a12, b12], [c12]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a12, b12, c12], [a12, b12], [[a12, b12, [[a12, c12], [b12]]], [a12, c12, [[a12, b12], [c12]]], [b12, c12, [[a12, b12], [c12]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123013,"parameters":{"points":["a12","b12","c12"],"topology":[["a12","b12","c12"],["a12","b12"],["a12","c12"],["a12"],["b12","c12"],["b12"],["c12"],[]],"compact_subset":["a12","b12"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a12\",\"b12\"],\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"b12\",\"c12\"],[\"b12\"],[\"c12\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0014","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"b13\",\"c13\"],[\"b13\"],[\"c13\"],[]],\"compact_subset\":[\"a13\",\"b13\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a13, b13, c13], [a13, b13], [[a13, b13, [[a13, c13], [b13]]], [a13, c13, [[a13, b13], [c13]]], [b13, c13, [[a13, b13], [c13]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a13, b13, c13], [a13, b13], [[a13, b13, [[a13, c13], [b13]]], [a13, c13, [[a13, b13], [c13]]], [b13, c13, [[a13, b13], [c13]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123014,"parameters":{"points":["a13","b13","c13"],"topology":[["a13","b13","c13"],["a13","b13"],["a13","c13"],["a13"],["b13","c13"],["b13"],["c13"],[]],"compact_subset":["a13","b13"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a13\",\"b13\"],\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"b13\",\"c13\"],[\"b13\"],[\"c13\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0015","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"b14\",\"c14\"],[\"b14\"],[\"c14\"],[]],\"compact_subset\":[\"a14\",\"b14\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a14, b14, c14], [a14, b14], [[a14, b14, [[a14, c14], [b14]]], [a14, c14, [[a14, b14], [c14]]], [b14, c14, [[a14, b14], [c14]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a14, b14, c14], [a14, b14], [[a14, b14, [[a14, c14], [b14]]], [a14, c14, [[a14, b14], [c14]]], [b14, c14, [[a14, b14], [c14]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123015,"parameters":{"points":["a14","b14","c14"],"topology":[["a14","b14","c14"],["a14","b14"],["a14","c14"],["a14"],["b14","c14"],["b14"],["c14"],[]],"compact_subset":["a14","b14"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a14\",\"b14\"],\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"b14\",\"c14\"],[\"b14\"],[\"c14\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0016","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"b15\",\"c15\"],[\"b15\"],[\"c15\"],[]],\"compact_subset\":[\"a15\",\"b15\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a15, b15, c15], [a15, b15], [[a15, b15, [[a15, c15], [b15]]], [a15, c15, [[a15, b15], [c15]]], [b15, c15, [[a15, b15], [c15]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a15, b15, c15], [a15, b15], [[a15, b15, [[a15, c15], [b15]]], [a15, c15, [[a15, b15], [c15]]], [b15, c15, [[a15, b15], [c15]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123016,"parameters":{"points":["a15","b15","c15"],"topology":[["a15","b15","c15"],["a15","b15"],["a15","c15"],["a15"],["b15","c15"],["b15"],["c15"],[]],"compact_subset":["a15","b15"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a15\",\"b15\"],\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"b15\",\"c15\"],[\"b15\"],[\"c15\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0017","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"b16\",\"c16\"],[\"b16\"],[\"c16\"],[]],\"compact_subset\":[\"a16\",\"b16\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a16, b16, c16], [a16, b16], [[a16, b16, [[a16, c16], [b16]]], [a16, c16, [[a16, b16], [c16]]], [b16, c16, [[a16, b16], [c16]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a16, b16, c16], [a16, b16], [[a16, b16, [[a16, c16], [b16]]], [a16, c16, [[a16, b16], [c16]]], [b16, c16, [[a16, b16], [c16]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123017,"parameters":{"points":["a16","b16","c16"],"topology":[["a16","b16","c16"],["a16","b16"],["a16","c16"],["a16"],["b16","c16"],["b16"],["c16"],[]],"compact_subset":["a16","b16"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a16\",\"b16\"],\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"b16\",\"c16\"],[\"b16\"],[\"c16\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0018","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"b17\",\"c17\"],[\"b17\"],[\"c17\"],[]],\"compact_subset\":[\"a17\",\"b17\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a17, b17, c17], [a17, b17], [[a17, b17, [[a17, c17], [b17]]], [a17, c17, [[a17, b17], [c17]]], [b17, c17, [[a17, b17], [c17]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a17, b17, c17], [a17, b17], [[a17, b17, [[a17, c17], [b17]]], [a17, c17, [[a17, b17], [c17]]], [b17, c17, [[a17, b17], [c17]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123018,"parameters":{"points":["a17","b17","c17"],"topology":[["a17","b17","c17"],["a17","b17"],["a17","c17"],["a17"],["b17","c17"],["b17"],["c17"],[]],"compact_subset":["a17","b17"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a17\",\"b17\"],\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"b17\",\"c17\"],[\"b17\"],[\"c17\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0019","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"b18\",\"c18\"],[\"b18\"],[\"c18\"],[]],\"compact_subset\":[\"a18\",\"b18\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a18, b18, c18], [a18, b18], [[a18, b18, [[a18, c18], [b18]]], [a18, c18, [[a18, b18], [c18]]], [b18, c18, [[a18, b18], [c18]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a18, b18, c18], [a18, b18], [[a18, b18, [[a18, c18], [b18]]], [a18, c18, [[a18, b18], [c18]]], [b18, c18, [[a18, b18], [c18]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123019,"parameters":{"points":["a18","b18","c18"],"topology":[["a18","b18","c18"],["a18","b18"],["a18","c18"],["a18"],["b18","c18"],["b18"],["c18"],[]],"compact_subset":["a18","b18"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a18\",\"b18\"],\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"b18\",\"c18\"],[\"b18\"],[\"c18\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0020","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"b19\",\"c19\"],[\"b19\"],[\"c19\"],[]],\"compact_subset\":[\"a19\",\"b19\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a19, b19, c19], [a19, b19], [[a19, b19, [[a19, c19], [b19]]], [a19, c19, [[a19, b19], [c19]]], [b19, c19, [[a19, b19], [c19]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a19, b19, c19], [a19, b19], [[a19, b19, [[a19, c19], [b19]]], [a19, c19, [[a19, b19], [c19]]], [b19, c19, [[a19, b19], [c19]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123020,"parameters":{"points":["a19","b19","c19"],"topology":[["a19","b19","c19"],["a19","b19"],["a19","c19"],["a19"],["b19","c19"],["b19"],["c19"],[]],"compact_subset":["a19","b19"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a19\",\"b19\"],\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"b19\",\"c19\"],[\"b19\"],[\"c19\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0021","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]],\"compact_subset\":[\"a20\",\"b20\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a20, b20, c20], [a20, b20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a20, b20, c20], [a20, b20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123021,"parameters":{"points":["a20","b20","c20"],"topology":[["a20","b20","c20"],["a20","b20"],["a20","c20"],["a20"],["b20","c20"],["b20"],["c20"],[]],"compact_subset":["a20","b20"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a20\",\"b20\"],\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0022","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"b21\",\"c21\"],[\"b21\"],[\"c21\"],[]],\"compact_subset\":[\"a21\",\"b21\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a21, b21, c21], [a21, b21], [[a21, b21, [[a21, c21], [b21]]], [a21, c21, [[a21, b21], [c21]]], [b21, c21, [[a21, b21], [c21]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a21, b21, c21], [a21, b21], [[a21, b21, [[a21, c21], [b21]]], [a21, c21, [[a21, b21], [c21]]], [b21, c21, [[a21, b21], [c21]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123022,"parameters":{"points":["a21","b21","c21"],"topology":[["a21","b21","c21"],["a21","b21"],["a21","c21"],["a21"],["b21","c21"],["b21"],["c21"],[]],"compact_subset":["a21","b21"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a21\",\"b21\"],\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"b21\",\"c21\"],[\"b21\"],[\"c21\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0023","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"b22\",\"c22\"],[\"b22\"],[\"c22\"],[]],\"compact_subset\":[\"a22\",\"b22\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a22, b22, c22], [a22, b22], [[a22, b22, [[a22, c22], [b22]]], [a22, c22, [[a22, b22], [c22]]], [b22, c22, [[a22, b22], [c22]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a22, b22, c22], [a22, b22], [[a22, b22, [[a22, c22], [b22]]], [a22, c22, [[a22, b22], [c22]]], [b22, c22, [[a22, b22], [c22]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123023,"parameters":{"points":["a22","b22","c22"],"topology":[["a22","b22","c22"],["a22","b22"],["a22","c22"],["a22"],["b22","c22"],["b22"],["c22"],[]],"compact_subset":["a22","b22"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a22\",\"b22\"],\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"b22\",\"c22\"],[\"b22\"],[\"c22\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0024","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"b23\",\"c23\"],[\"b23\"],[\"c23\"],[]],\"compact_subset\":[\"a23\",\"b23\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a23, b23, c23], [a23, b23], [[a23, b23, [[a23, c23], [b23]]], [a23, c23, [[a23, b23], [c23]]], [b23, c23, [[a23, b23], [c23]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a23, b23, c23], [a23, b23], [[a23, b23, [[a23, c23], [b23]]], [a23, c23, [[a23, b23], [c23]]], [b23, c23, [[a23, b23], [c23]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123024,"parameters":{"points":["a23","b23","c23"],"topology":[["a23","b23","c23"],["a23","b23"],["a23","c23"],["a23"],["b23","c23"],["b23"],["c23"],[]],"compact_subset":["a23","b23"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a23\",\"b23\"],\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"b23\",\"c23\"],[\"b23\"],[\"c23\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0025","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"b24\",\"c24\"],[\"b24\"],[\"c24\"],[]],\"compact_subset\":[\"a24\",\"b24\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a24, b24, c24], [a24, b24], [[a24, b24, [[a24, c24], [b24]]], [a24, c24, [[a24, b24], [c24]]], [b24, c24, [[a24, b24], [c24]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)","explanation_zh":"Wave6第六層核對=([a24, b24, c24], [a24, b24], [[a24, b24, [[a24, c24], [b24]]], [a24, c24, [[a24, b24], [c24]]], [b24, c24, [[a24, b24], [c24]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-hypotheses-qualification","seed":202611123025,"parameters":{"points":["a24","b24","c24"],"topology":[["a24","b24","c24"],["a24","b24"],["a24","c24"],["a24"],["b24","c24"],["b24"],["c24"],[]],"compact_subset":["a24","b24"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-hypotheses-qualification","semantic_template":"qf-mod43-compactclosed-hypotheses-qualification/{\"compact_subset\":[\"a24\",\"b24\"],\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"b24\",\"c24\"],[\"b24\"],[\"c24\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0026","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"b0\",\"c0\"],[\"b0\"],[\"c0\"],[]],\"compact_subset\":[\"a0\",\"b0\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, b0], [c0], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a0, b0], [c0], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123026,"parameters":{"points":["a0","b0","c0"],"topology":[["a0","b0","c0"],["a0","b0"],["a0","c0"],["a0"],["b0","c0"],["b0"],["c0"],[]],"compact_subset":["a0","b0"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a0\",\"b0\"],\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"b0\",\"c0\"],[\"b0\"],[\"c0\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0027","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"b1\",\"c1\"],[\"b1\"],[\"c1\"],[]],\"compact_subset\":[\"a1\",\"b1\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a1, b1], [c1], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a1, b1], [c1], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123027,"parameters":{"points":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["b1","c1"],["b1"],["c1"],[]],"compact_subset":["a1","b1"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a1\",\"b1\"],\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"b1\",\"c1\"],[\"b1\"],[\"c1\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0028","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"b2\",\"c2\"],[\"b2\"],[\"c2\"],[]],\"compact_subset\":[\"a2\",\"b2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a2, b2], [c2], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a2, b2], [c2], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123028,"parameters":{"points":["a2","b2","c2"],"topology":[["a2","b2","c2"],["a2","b2"],["a2","c2"],["a2"],["b2","c2"],["b2"],["c2"],[]],"compact_subset":["a2","b2"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a2\",\"b2\"],\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"b2\",\"c2\"],[\"b2\"],[\"c2\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0029","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"b3\",\"c3\"],[\"b3\"],[\"c3\"],[]],\"compact_subset\":[\"a3\",\"b3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a3, b3], [c3], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a3, b3], [c3], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123029,"parameters":{"points":["a3","b3","c3"],"topology":[["a3","b3","c3"],["a3","b3"],["a3","c3"],["a3"],["b3","c3"],["b3"],["c3"],[]],"compact_subset":["a3","b3"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a3\",\"b3\"],\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"b3\",\"c3\"],[\"b3\"],[\"c3\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0030","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"b4\",\"c4\"],[\"b4\"],[\"c4\"],[]],\"compact_subset\":[\"a4\",\"b4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a4, b4], [c4], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a4, b4], [c4], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123030,"parameters":{"points":["a4","b4","c4"],"topology":[["a4","b4","c4"],["a4","b4"],["a4","c4"],["a4"],["b4","c4"],["b4"],["c4"],[]],"compact_subset":["a4","b4"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a4\",\"b4\"],\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"b4\",\"c4\"],[\"b4\"],[\"c4\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0031","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"b5\",\"c5\"],[\"b5\"],[\"c5\"],[]],\"compact_subset\":[\"a5\",\"b5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a5, b5], [c5], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a5, b5], [c5], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123031,"parameters":{"points":["a5","b5","c5"],"topology":[["a5","b5","c5"],["a5","b5"],["a5","c5"],["a5"],["b5","c5"],["b5"],["c5"],[]],"compact_subset":["a5","b5"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a5\",\"b5\"],\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"b5\",\"c5\"],[\"b5\"],[\"c5\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0032","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"b6\",\"c6\"],[\"b6\"],[\"c6\"],[]],\"compact_subset\":[\"a6\",\"b6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a6, b6], [c6], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a6, b6], [c6], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123032,"parameters":{"points":["a6","b6","c6"],"topology":[["a6","b6","c6"],["a6","b6"],["a6","c6"],["a6"],["b6","c6"],["b6"],["c6"],[]],"compact_subset":["a6","b6"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a6\",\"b6\"],\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"b6\",\"c6\"],[\"b6\"],[\"c6\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0033","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"b7\",\"c7\"],[\"b7\"],[\"c7\"],[]],\"compact_subset\":[\"a7\",\"b7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a7, b7], [c7], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a7, b7], [c7], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123033,"parameters":{"points":["a7","b7","c7"],"topology":[["a7","b7","c7"],["a7","b7"],["a7","c7"],["a7"],["b7","c7"],["b7"],["c7"],[]],"compact_subset":["a7","b7"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a7\",\"b7\"],\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"b7\",\"c7\"],[\"b7\"],[\"c7\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0034","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"b8\",\"c8\"],[\"b8\"],[\"c8\"],[]],\"compact_subset\":[\"a8\",\"b8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a8, b8], [c8], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a8, b8], [c8], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123034,"parameters":{"points":["a8","b8","c8"],"topology":[["a8","b8","c8"],["a8","b8"],["a8","c8"],["a8"],["b8","c8"],["b8"],["c8"],[]],"compact_subset":["a8","b8"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a8\",\"b8\"],\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"b8\",\"c8\"],[\"b8\"],[\"c8\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0035","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"b9\",\"c9\"],[\"b9\"],[\"c9\"],[]],\"compact_subset\":[\"a9\",\"b9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a9, b9], [c9], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a9, b9], [c9], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123035,"parameters":{"points":["a9","b9","c9"],"topology":[["a9","b9","c9"],["a9","b9"],["a9","c9"],["a9"],["b9","c9"],["b9"],["c9"],[]],"compact_subset":["a9","b9"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a9\",\"b9\"],\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"b9\",\"c9\"],[\"b9\"],[\"c9\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0036","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"b10\",\"c10\"],[\"b10\"],[\"c10\"],[]],\"compact_subset\":[\"a10\",\"b10\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a10, b10], [c10], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a10, b10], [c10], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123036,"parameters":{"points":["a10","b10","c10"],"topology":[["a10","b10","c10"],["a10","b10"],["a10","c10"],["a10"],["b10","c10"],["b10"],["c10"],[]],"compact_subset":["a10","b10"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a10\",\"b10\"],\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"b10\",\"c10\"],[\"b10\"],[\"c10\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0037","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"b11\",\"c11\"],[\"b11\"],[\"c11\"],[]],\"compact_subset\":[\"a11\",\"b11\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a11, b11], [c11], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a11, b11], [c11], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123037,"parameters":{"points":["a11","b11","c11"],"topology":[["a11","b11","c11"],["a11","b11"],["a11","c11"],["a11"],["b11","c11"],["b11"],["c11"],[]],"compact_subset":["a11","b11"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a11\",\"b11\"],\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"b11\",\"c11\"],[\"b11\"],[\"c11\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0038","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"b12\",\"c12\"],[\"b12\"],[\"c12\"],[]],\"compact_subset\":[\"a12\",\"b12\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a12, b12], [c12], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a12, b12], [c12], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123038,"parameters":{"points":["a12","b12","c12"],"topology":[["a12","b12","c12"],["a12","b12"],["a12","c12"],["a12"],["b12","c12"],["b12"],["c12"],[]],"compact_subset":["a12","b12"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a12\",\"b12\"],\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"b12\",\"c12\"],[\"b12\"],[\"c12\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0039","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"b13\",\"c13\"],[\"b13\"],[\"c13\"],[]],\"compact_subset\":[\"a13\",\"b13\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a13, b13], [c13], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a13, b13], [c13], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123039,"parameters":{"points":["a13","b13","c13"],"topology":[["a13","b13","c13"],["a13","b13"],["a13","c13"],["a13"],["b13","c13"],["b13"],["c13"],[]],"compact_subset":["a13","b13"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a13\",\"b13\"],\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"b13\",\"c13\"],[\"b13\"],[\"c13\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0040","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"b14\",\"c14\"],[\"b14\"],[\"c14\"],[]],\"compact_subset\":[\"a14\",\"b14\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a14, b14], [c14], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a14, b14], [c14], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123040,"parameters":{"points":["a14","b14","c14"],"topology":[["a14","b14","c14"],["a14","b14"],["a14","c14"],["a14"],["b14","c14"],["b14"],["c14"],[]],"compact_subset":["a14","b14"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a14\",\"b14\"],\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"b14\",\"c14\"],[\"b14\"],[\"c14\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0041","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"b15\",\"c15\"],[\"b15\"],[\"c15\"],[]],\"compact_subset\":[\"a15\",\"b15\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a15, b15], [c15], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a15, b15], [c15], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123041,"parameters":{"points":["a15","b15","c15"],"topology":[["a15","b15","c15"],["a15","b15"],["a15","c15"],["a15"],["b15","c15"],["b15"],["c15"],[]],"compact_subset":["a15","b15"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a15\",\"b15\"],\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"b15\",\"c15\"],[\"b15\"],[\"c15\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0042","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"b16\",\"c16\"],[\"b16\"],[\"c16\"],[]],\"compact_subset\":[\"a16\",\"b16\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a16, b16], [c16], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a16, b16], [c16], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123042,"parameters":{"points":["a16","b16","c16"],"topology":[["a16","b16","c16"],["a16","b16"],["a16","c16"],["a16"],["b16","c16"],["b16"],["c16"],[]],"compact_subset":["a16","b16"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a16\",\"b16\"],\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"b16\",\"c16\"],[\"b16\"],[\"c16\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0043","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"b17\",\"c17\"],[\"b17\"],[\"c17\"],[]],\"compact_subset\":[\"a17\",\"b17\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a17, b17], [c17], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a17, b17], [c17], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123043,"parameters":{"points":["a17","b17","c17"],"topology":[["a17","b17","c17"],["a17","b17"],["a17","c17"],["a17"],["b17","c17"],["b17"],["c17"],[]],"compact_subset":["a17","b17"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a17\",\"b17\"],\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"b17\",\"c17\"],[\"b17\"],[\"c17\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0044","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"b18\",\"c18\"],[\"b18\"],[\"c18\"],[]],\"compact_subset\":[\"a18\",\"b18\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a18, b18], [c18], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a18, b18], [c18], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123044,"parameters":{"points":["a18","b18","c18"],"topology":[["a18","b18","c18"],["a18","b18"],["a18","c18"],["a18"],["b18","c18"],["b18"],["c18"],[]],"compact_subset":["a18","b18"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a18\",\"b18\"],\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"b18\",\"c18\"],[\"b18\"],[\"c18\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0045","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"b19\",\"c19\"],[\"b19\"],[\"c19\"],[]],\"compact_subset\":[\"a19\",\"b19\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a19, b19], [c19], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a19, b19], [c19], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123045,"parameters":{"points":["a19","b19","c19"],"topology":[["a19","b19","c19"],["a19","b19"],["a19","c19"],["a19"],["b19","c19"],["b19"],["c19"],[]],"compact_subset":["a19","b19"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a19\",\"b19\"],\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"b19\",\"c19\"],[\"b19\"],[\"c19\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0046","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]],\"compact_subset\":[\"a20\",\"b20\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a20, b20], [c20], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a20, b20], [c20], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123046,"parameters":{"points":["a20","b20","c20"],"topology":[["a20","b20","c20"],["a20","b20"],["a20","c20"],["a20"],["b20","c20"],["b20"],["c20"],[]],"compact_subset":["a20","b20"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a20\",\"b20\"],\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0047","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"b21\",\"c21\"],[\"b21\"],[\"c21\"],[]],\"compact_subset\":[\"a21\",\"b21\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a21, b21], [c21], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a21, b21], [c21], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123047,"parameters":{"points":["a21","b21","c21"],"topology":[["a21","b21","c21"],["a21","b21"],["a21","c21"],["a21"],["b21","c21"],["b21"],["c21"],[]],"compact_subset":["a21","b21"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a21\",\"b21\"],\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"b21\",\"c21\"],[\"b21\"],[\"c21\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0048","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"b22\",\"c22\"],[\"b22\"],[\"c22\"],[]],\"compact_subset\":[\"a22\",\"b22\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a22, b22], [c22], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a22, b22], [c22], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123048,"parameters":{"points":["a22","b22","c22"],"topology":[["a22","b22","c22"],["a22","b22"],["a22","c22"],["a22"],["b22","c22"],["b22"],["c22"],[]],"compact_subset":["a22","b22"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a22\",\"b22\"],\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"b22\",\"c22\"],[\"b22\"],[\"c22\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0049","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"b23\",\"c23\"],[\"b23\"],[\"c23\"],[]],\"compact_subset\":[\"a23\",\"b23\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a23, b23], [c23], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a23, b23], [c23], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123049,"parameters":{"points":["a23","b23","c23"],"topology":[["a23","b23","c23"],["a23","b23"],["a23","c23"],["a23"],["b23","c23"],["b23"],["c23"],[]],"compact_subset":["a23","b23"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a23\",\"b23\"],\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"b23\",\"c23\"],[\"b23\"],[\"c23\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0050","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"b24\",\"c24\"],[\"b24\"],[\"c24\"],[]],\"compact_subset\":[\"a24\",\"b24\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a24, b24], [c24], 是, 是, complement_is_open_so_compact_subset_is_closed)","explanation_zh":"Wave6第六層核對=([a24, b24], [c24], 是, 是, complement_is_open_so_compact_subset_is_closed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-finite-closedness-audit","seed":202611123050,"parameters":{"points":["a24","b24","c24"],"topology":[["a24","b24","c24"],["a24","b24"],["a24","c24"],["a24"],["b24","c24"],["b24"],["c24"],[]],"compact_subset":["a24","b24"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-finite-closedness-audit","semantic_template":"qf-mod43-compactclosed-finite-closedness-audit/{\"compact_subset\":[\"a24\",\"b24\"],\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"b24\",\"c24\"],[\"b24\"],[\"c24\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0051","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"b0\",\"c0\"],[\"b0\"],[\"c0\"],[]],\"compact_subset\":[\"a0\",\"b0\"],\"exterior_point\":\"c0\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c0, [[a0, [[b0, c0], [a0]]], [b0, [[a0, c0], [b0]]]], [c0], [a0, b0], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c0, [[a0, [[b0, c0], [a0]]], [b0, [[a0, c0], [b0]]]], [c0], [a0, b0], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123051,"parameters":{"points":["a0","b0","c0"],"topology":[["a0","b0","c0"],["a0","b0"],["a0","c0"],["a0"],["b0","c0"],["b0"],["c0"],[]],"compact_subset":["a0","b0"],"exterior_point":"c0","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a0\",\"b0\"],\"exterior_point\":\"c0\",\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"b0\",\"c0\"],[\"b0\"],[\"c0\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0052","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"b1\",\"c1\"],[\"b1\"],[\"c1\"],[]],\"compact_subset\":[\"a1\",\"b1\"],\"exterior_point\":\"c1\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c1, [[a1, [[b1, c1], [a1]]], [b1, [[a1, c1], [b1]]]], [c1], [a1, b1], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c1, [[a1, [[b1, c1], [a1]]], [b1, [[a1, c1], [b1]]]], [c1], [a1, b1], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123052,"parameters":{"points":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["b1","c1"],["b1"],["c1"],[]],"compact_subset":["a1","b1"],"exterior_point":"c1","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a1\",\"b1\"],\"exterior_point\":\"c1\",\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"b1\",\"c1\"],[\"b1\"],[\"c1\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0053","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"b2\",\"c2\"],[\"b2\"],[\"c2\"],[]],\"compact_subset\":[\"a2\",\"b2\"],\"exterior_point\":\"c2\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c2, [[a2, [[b2, c2], [a2]]], [b2, [[a2, c2], [b2]]]], [c2], [a2, b2], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c2, [[a2, [[b2, c2], [a2]]], [b2, [[a2, c2], [b2]]]], [c2], [a2, b2], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123053,"parameters":{"points":["a2","b2","c2"],"topology":[["a2","b2","c2"],["a2","b2"],["a2","c2"],["a2"],["b2","c2"],["b2"],["c2"],[]],"compact_subset":["a2","b2"],"exterior_point":"c2","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a2\",\"b2\"],\"exterior_point\":\"c2\",\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"b2\",\"c2\"],[\"b2\"],[\"c2\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0054","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"b3\",\"c3\"],[\"b3\"],[\"c3\"],[]],\"compact_subset\":[\"a3\",\"b3\"],\"exterior_point\":\"c3\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c3, [[a3, [[b3, c3], [a3]]], [b3, [[a3, c3], [b3]]]], [c3], [a3, b3], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c3, [[a3, [[b3, c3], [a3]]], [b3, [[a3, c3], [b3]]]], [c3], [a3, b3], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123054,"parameters":{"points":["a3","b3","c3"],"topology":[["a3","b3","c3"],["a3","b3"],["a3","c3"],["a3"],["b3","c3"],["b3"],["c3"],[]],"compact_subset":["a3","b3"],"exterior_point":"c3","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a3\",\"b3\"],\"exterior_point\":\"c3\",\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"b3\",\"c3\"],[\"b3\"],[\"c3\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0055","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"b4\",\"c4\"],[\"b4\"],[\"c4\"],[]],\"compact_subset\":[\"a4\",\"b4\"],\"exterior_point\":\"c4\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c4, [[a4, [[b4, c4], [a4]]], [b4, [[a4, c4], [b4]]]], [c4], [a4, b4], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c4, [[a4, [[b4, c4], [a4]]], [b4, [[a4, c4], [b4]]]], [c4], [a4, b4], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123055,"parameters":{"points":["a4","b4","c4"],"topology":[["a4","b4","c4"],["a4","b4"],["a4","c4"],["a4"],["b4","c4"],["b4"],["c4"],[]],"compact_subset":["a4","b4"],"exterior_point":"c4","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a4\",\"b4\"],\"exterior_point\":\"c4\",\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"b4\",\"c4\"],[\"b4\"],[\"c4\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0056","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"b5\",\"c5\"],[\"b5\"],[\"c5\"],[]],\"compact_subset\":[\"a5\",\"b5\"],\"exterior_point\":\"c5\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c5, [[a5, [[b5, c5], [a5]]], [b5, [[a5, c5], [b5]]]], [c5], [a5, b5], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c5, [[a5, [[b5, c5], [a5]]], [b5, [[a5, c5], [b5]]]], [c5], [a5, b5], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123056,"parameters":{"points":["a5","b5","c5"],"topology":[["a5","b5","c5"],["a5","b5"],["a5","c5"],["a5"],["b5","c5"],["b5"],["c5"],[]],"compact_subset":["a5","b5"],"exterior_point":"c5","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a5\",\"b5\"],\"exterior_point\":\"c5\",\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"b5\",\"c5\"],[\"b5\"],[\"c5\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0057","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"b6\",\"c6\"],[\"b6\"],[\"c6\"],[]],\"compact_subset\":[\"a6\",\"b6\"],\"exterior_point\":\"c6\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c6, [[a6, [[b6, c6], [a6]]], [b6, [[a6, c6], [b6]]]], [c6], [a6, b6], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c6, [[a6, [[b6, c6], [a6]]], [b6, [[a6, c6], [b6]]]], [c6], [a6, b6], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123057,"parameters":{"points":["a6","b6","c6"],"topology":[["a6","b6","c6"],["a6","b6"],["a6","c6"],["a6"],["b6","c6"],["b6"],["c6"],[]],"compact_subset":["a6","b6"],"exterior_point":"c6","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a6\",\"b6\"],\"exterior_point\":\"c6\",\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"b6\",\"c6\"],[\"b6\"],[\"c6\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0058","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"b7\",\"c7\"],[\"b7\"],[\"c7\"],[]],\"compact_subset\":[\"a7\",\"b7\"],\"exterior_point\":\"c7\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c7, [[a7, [[b7, c7], [a7]]], [b7, [[a7, c7], [b7]]]], [c7], [a7, b7], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c7, [[a7, [[b7, c7], [a7]]], [b7, [[a7, c7], [b7]]]], [c7], [a7, b7], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123058,"parameters":{"points":["a7","b7","c7"],"topology":[["a7","b7","c7"],["a7","b7"],["a7","c7"],["a7"],["b7","c7"],["b7"],["c7"],[]],"compact_subset":["a7","b7"],"exterior_point":"c7","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a7\",\"b7\"],\"exterior_point\":\"c7\",\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"b7\",\"c7\"],[\"b7\"],[\"c7\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0059","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"b8\",\"c8\"],[\"b8\"],[\"c8\"],[]],\"compact_subset\":[\"a8\",\"b8\"],\"exterior_point\":\"c8\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c8, [[a8, [[b8, c8], [a8]]], [b8, [[a8, c8], [b8]]]], [c8], [a8, b8], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c8, [[a8, [[b8, c8], [a8]]], [b8, [[a8, c8], [b8]]]], [c8], [a8, b8], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123059,"parameters":{"points":["a8","b8","c8"],"topology":[["a8","b8","c8"],["a8","b8"],["a8","c8"],["a8"],["b8","c8"],["b8"],["c8"],[]],"compact_subset":["a8","b8"],"exterior_point":"c8","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a8\",\"b8\"],\"exterior_point\":\"c8\",\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"b8\",\"c8\"],[\"b8\"],[\"c8\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0060","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"b9\",\"c9\"],[\"b9\"],[\"c9\"],[]],\"compact_subset\":[\"a9\",\"b9\"],\"exterior_point\":\"c9\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c9, [[a9, [[b9, c9], [a9]]], [b9, [[a9, c9], [b9]]]], [c9], [a9, b9], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c9, [[a9, [[b9, c9], [a9]]], [b9, [[a9, c9], [b9]]]], [c9], [a9, b9], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123060,"parameters":{"points":["a9","b9","c9"],"topology":[["a9","b9","c9"],["a9","b9"],["a9","c9"],["a9"],["b9","c9"],["b9"],["c9"],[]],"compact_subset":["a9","b9"],"exterior_point":"c9","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a9\",\"b9\"],\"exterior_point\":\"c9\",\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"b9\",\"c9\"],[\"b9\"],[\"c9\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0061","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"b10\",\"c10\"],[\"b10\"],[\"c10\"],[]],\"compact_subset\":[\"a10\",\"b10\"],\"exterior_point\":\"c10\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c10, [[a10, [[b10, c10], [a10]]], [b10, [[a10, c10], [b10]]]], [c10], [a10, b10], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c10, [[a10, [[b10, c10], [a10]]], [b10, [[a10, c10], [b10]]]], [c10], [a10, b10], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123061,"parameters":{"points":["a10","b10","c10"],"topology":[["a10","b10","c10"],["a10","b10"],["a10","c10"],["a10"],["b10","c10"],["b10"],["c10"],[]],"compact_subset":["a10","b10"],"exterior_point":"c10","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a10\",\"b10\"],\"exterior_point\":\"c10\",\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"b10\",\"c10\"],[\"b10\"],[\"c10\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0062","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"b11\",\"c11\"],[\"b11\"],[\"c11\"],[]],\"compact_subset\":[\"a11\",\"b11\"],\"exterior_point\":\"c11\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c11, [[a11, [[b11, c11], [a11]]], [b11, [[a11, c11], [b11]]]], [c11], [a11, b11], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c11, [[a11, [[b11, c11], [a11]]], [b11, [[a11, c11], [b11]]]], [c11], [a11, b11], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123062,"parameters":{"points":["a11","b11","c11"],"topology":[["a11","b11","c11"],["a11","b11"],["a11","c11"],["a11"],["b11","c11"],["b11"],["c11"],[]],"compact_subset":["a11","b11"],"exterior_point":"c11","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a11\",\"b11\"],\"exterior_point\":\"c11\",\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"b11\",\"c11\"],[\"b11\"],[\"c11\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0063","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"b12\",\"c12\"],[\"b12\"],[\"c12\"],[]],\"compact_subset\":[\"a12\",\"b12\"],\"exterior_point\":\"c12\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c12, [[a12, [[b12, c12], [a12]]], [b12, [[a12, c12], [b12]]]], [c12], [a12, b12], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c12, [[a12, [[b12, c12], [a12]]], [b12, [[a12, c12], [b12]]]], [c12], [a12, b12], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123063,"parameters":{"points":["a12","b12","c12"],"topology":[["a12","b12","c12"],["a12","b12"],["a12","c12"],["a12"],["b12","c12"],["b12"],["c12"],[]],"compact_subset":["a12","b12"],"exterior_point":"c12","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a12\",\"b12\"],\"exterior_point\":\"c12\",\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"b12\",\"c12\"],[\"b12\"],[\"c12\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0064","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"b13\",\"c13\"],[\"b13\"],[\"c13\"],[]],\"compact_subset\":[\"a13\",\"b13\"],\"exterior_point\":\"c13\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c13, [[a13, [[b13, c13], [a13]]], [b13, [[a13, c13], [b13]]]], [c13], [a13, b13], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c13, [[a13, [[b13, c13], [a13]]], [b13, [[a13, c13], [b13]]]], [c13], [a13, b13], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123064,"parameters":{"points":["a13","b13","c13"],"topology":[["a13","b13","c13"],["a13","b13"],["a13","c13"],["a13"],["b13","c13"],["b13"],["c13"],[]],"compact_subset":["a13","b13"],"exterior_point":"c13","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a13\",\"b13\"],\"exterior_point\":\"c13\",\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"b13\",\"c13\"],[\"b13\"],[\"c13\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0065","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"b14\",\"c14\"],[\"b14\"],[\"c14\"],[]],\"compact_subset\":[\"a14\",\"b14\"],\"exterior_point\":\"c14\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c14, [[a14, [[b14, c14], [a14]]], [b14, [[a14, c14], [b14]]]], [c14], [a14, b14], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c14, [[a14, [[b14, c14], [a14]]], [b14, [[a14, c14], [b14]]]], [c14], [a14, b14], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123065,"parameters":{"points":["a14","b14","c14"],"topology":[["a14","b14","c14"],["a14","b14"],["a14","c14"],["a14"],["b14","c14"],["b14"],["c14"],[]],"compact_subset":["a14","b14"],"exterior_point":"c14","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a14\",\"b14\"],\"exterior_point\":\"c14\",\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"b14\",\"c14\"],[\"b14\"],[\"c14\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0066","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"b15\",\"c15\"],[\"b15\"],[\"c15\"],[]],\"compact_subset\":[\"a15\",\"b15\"],\"exterior_point\":\"c15\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c15, [[a15, [[b15, c15], [a15]]], [b15, [[a15, c15], [b15]]]], [c15], [a15, b15], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c15, [[a15, [[b15, c15], [a15]]], [b15, [[a15, c15], [b15]]]], [c15], [a15, b15], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123066,"parameters":{"points":["a15","b15","c15"],"topology":[["a15","b15","c15"],["a15","b15"],["a15","c15"],["a15"],["b15","c15"],["b15"],["c15"],[]],"compact_subset":["a15","b15"],"exterior_point":"c15","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a15\",\"b15\"],\"exterior_point\":\"c15\",\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"b15\",\"c15\"],[\"b15\"],[\"c15\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0067","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"b16\",\"c16\"],[\"b16\"],[\"c16\"],[]],\"compact_subset\":[\"a16\",\"b16\"],\"exterior_point\":\"c16\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c16, [[a16, [[b16, c16], [a16]]], [b16, [[a16, c16], [b16]]]], [c16], [a16, b16], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c16, [[a16, [[b16, c16], [a16]]], [b16, [[a16, c16], [b16]]]], [c16], [a16, b16], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123067,"parameters":{"points":["a16","b16","c16"],"topology":[["a16","b16","c16"],["a16","b16"],["a16","c16"],["a16"],["b16","c16"],["b16"],["c16"],[]],"compact_subset":["a16","b16"],"exterior_point":"c16","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a16\",\"b16\"],\"exterior_point\":\"c16\",\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"b16\",\"c16\"],[\"b16\"],[\"c16\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0068","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"b17\",\"c17\"],[\"b17\"],[\"c17\"],[]],\"compact_subset\":[\"a17\",\"b17\"],\"exterior_point\":\"c17\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c17, [[a17, [[b17, c17], [a17]]], [b17, [[a17, c17], [b17]]]], [c17], [a17, b17], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c17, [[a17, [[b17, c17], [a17]]], [b17, [[a17, c17], [b17]]]], [c17], [a17, b17], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123068,"parameters":{"points":["a17","b17","c17"],"topology":[["a17","b17","c17"],["a17","b17"],["a17","c17"],["a17"],["b17","c17"],["b17"],["c17"],[]],"compact_subset":["a17","b17"],"exterior_point":"c17","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a17\",\"b17\"],\"exterior_point\":\"c17\",\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"b17\",\"c17\"],[\"b17\"],[\"c17\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0069","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"b18\",\"c18\"],[\"b18\"],[\"c18\"],[]],\"compact_subset\":[\"a18\",\"b18\"],\"exterior_point\":\"c18\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c18, [[a18, [[b18, c18], [a18]]], [b18, [[a18, c18], [b18]]]], [c18], [a18, b18], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c18, [[a18, [[b18, c18], [a18]]], [b18, [[a18, c18], [b18]]]], [c18], [a18, b18], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123069,"parameters":{"points":["a18","b18","c18"],"topology":[["a18","b18","c18"],["a18","b18"],["a18","c18"],["a18"],["b18","c18"],["b18"],["c18"],[]],"compact_subset":["a18","b18"],"exterior_point":"c18","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a18\",\"b18\"],\"exterior_point\":\"c18\",\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"b18\",\"c18\"],[\"b18\"],[\"c18\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0070","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"b19\",\"c19\"],[\"b19\"],[\"c19\"],[]],\"compact_subset\":[\"a19\",\"b19\"],\"exterior_point\":\"c19\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c19, [[a19, [[b19, c19], [a19]]], [b19, [[a19, c19], [b19]]]], [c19], [a19, b19], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c19, [[a19, [[b19, c19], [a19]]], [b19, [[a19, c19], [b19]]]], [c19], [a19, b19], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123070,"parameters":{"points":["a19","b19","c19"],"topology":[["a19","b19","c19"],["a19","b19"],["a19","c19"],["a19"],["b19","c19"],["b19"],["c19"],[]],"compact_subset":["a19","b19"],"exterior_point":"c19","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a19\",\"b19\"],\"exterior_point\":\"c19\",\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"b19\",\"c19\"],[\"b19\"],[\"c19\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0071","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]],\"compact_subset\":[\"a20\",\"b20\"],\"exterior_point\":\"c20\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c20, [[a20, [[b20, c20], [a20]]], [b20, [[a20, c20], [b20]]]], [c20], [a20, b20], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c20, [[a20, [[b20, c20], [a20]]], [b20, [[a20, c20], [b20]]]], [c20], [a20, b20], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123071,"parameters":{"points":["a20","b20","c20"],"topology":[["a20","b20","c20"],["a20","b20"],["a20","c20"],["a20"],["b20","c20"],["b20"],["c20"],[]],"compact_subset":["a20","b20"],"exterior_point":"c20","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a20\",\"b20\"],\"exterior_point\":\"c20\",\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0072","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"b21\",\"c21\"],[\"b21\"],[\"c21\"],[]],\"compact_subset\":[\"a21\",\"b21\"],\"exterior_point\":\"c21\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c21, [[a21, [[b21, c21], [a21]]], [b21, [[a21, c21], [b21]]]], [c21], [a21, b21], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c21, [[a21, [[b21, c21], [a21]]], [b21, [[a21, c21], [b21]]]], [c21], [a21, b21], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123072,"parameters":{"points":["a21","b21","c21"],"topology":[["a21","b21","c21"],["a21","b21"],["a21","c21"],["a21"],["b21","c21"],["b21"],["c21"],[]],"compact_subset":["a21","b21"],"exterior_point":"c21","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a21\",\"b21\"],\"exterior_point\":\"c21\",\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"b21\",\"c21\"],[\"b21\"],[\"c21\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0073","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"b22\",\"c22\"],[\"b22\"],[\"c22\"],[]],\"compact_subset\":[\"a22\",\"b22\"],\"exterior_point\":\"c22\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c22, [[a22, [[b22, c22], [a22]]], [b22, [[a22, c22], [b22]]]], [c22], [a22, b22], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c22, [[a22, [[b22, c22], [a22]]], [b22, [[a22, c22], [b22]]]], [c22], [a22, b22], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123073,"parameters":{"points":["a22","b22","c22"],"topology":[["a22","b22","c22"],["a22","b22"],["a22","c22"],["a22"],["b22","c22"],["b22"],["c22"],[]],"compact_subset":["a22","b22"],"exterior_point":"c22","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a22\",\"b22\"],\"exterior_point\":\"c22\",\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"b22\",\"c22\"],[\"b22\"],[\"c22\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0074","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"b23\",\"c23\"],[\"b23\"],[\"c23\"],[]],\"compact_subset\":[\"a23\",\"b23\"],\"exterior_point\":\"c23\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c23, [[a23, [[b23, c23], [a23]]], [b23, [[a23, c23], [b23]]]], [c23], [a23, b23], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c23, [[a23, [[b23, c23], [a23]]], [b23, [[a23, c23], [b23]]]], [c23], [a23, b23], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123074,"parameters":{"points":["a23","b23","c23"],"topology":[["a23","b23","c23"],["a23","b23"],["a23","c23"],["a23"],["b23","c23"],["b23"],["c23"],[]],"compact_subset":["a23","b23"],"exterior_point":"c23","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a23\",\"b23\"],\"exterior_point\":\"c23\",\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"b23\",\"c23\"],[\"b23\"],[\"c23\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0075","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"b24\",\"c24\"],[\"b24\"],[\"c24\"],[]],\"compact_subset\":[\"a24\",\"b24\"],\"exterior_point\":\"c24\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(c24, [[a24, [[b24, c24], [a24]]], [b24, [[a24, c24], [b24]]]], [c24], [a24, b24], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)","explanation_zh":"Wave6第六層核對=(c24, [[a24, [[b24, c24], [a24]]], [b24, [[a24, c24], [b24]]]], [c24], [a24, b24], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-neighborhood-subcover-certificate","seed":202611123075,"parameters":{"points":["a24","b24","c24"],"topology":[["a24","b24","c24"],["a24","b24"],["a24","c24"],["a24"],["b24","c24"],["b24"],["c24"],[]],"compact_subset":["a24","b24"],"exterior_point":"c24","domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-neighborhood-subcover-certificate","semantic_template":"qf-mod43-compactclosed-neighborhood-subcover-certificate/{\"compact_subset\":[\"a24\",\"b24\"],\"exterior_point\":\"c24\",\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"b24\",\"c24\"],[\"b24\"],[\"c24\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0076","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o0\",\"u0\"],\"topology\":[[\"o0\",\"u0\"],[\"u0\"],[]],\"compact_subset\":[\"u0\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o0, u0], [u0], [o0], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o0, u0], [u0], [o0], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123076,"parameters":{"points":["o0","u0"],"topology":[["o0","u0"],["u0"],[]],"compact_subset":["u0"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u0\"],\"points\":[\"o0\",\"u0\"],\"topology\":[[\"o0\",\"u0\"],[\"u0\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0077","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o1\",\"u1\"],\"topology\":[[\"o1\",\"u1\"],[\"u1\"],[]],\"compact_subset\":[\"u1\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o1, u1], [u1], [o1], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o1, u1], [u1], [o1], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123077,"parameters":{"points":["o1","u1"],"topology":[["o1","u1"],["u1"],[]],"compact_subset":["u1"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u1\"],\"points\":[\"o1\",\"u1\"],\"topology\":[[\"o1\",\"u1\"],[\"u1\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0078","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o2\",\"u2\"],\"topology\":[[\"o2\",\"u2\"],[\"u2\"],[]],\"compact_subset\":[\"u2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o2, u2], [u2], [o2], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o2, u2], [u2], [o2], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123078,"parameters":{"points":["o2","u2"],"topology":[["o2","u2"],["u2"],[]],"compact_subset":["u2"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u2\"],\"points\":[\"o2\",\"u2\"],\"topology\":[[\"o2\",\"u2\"],[\"u2\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0079","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o3\",\"u3\"],\"topology\":[[\"o3\",\"u3\"],[\"u3\"],[]],\"compact_subset\":[\"u3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o3, u3], [u3], [o3], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o3, u3], [u3], [o3], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123079,"parameters":{"points":["o3","u3"],"topology":[["o3","u3"],["u3"],[]],"compact_subset":["u3"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u3\"],\"points\":[\"o3\",\"u3\"],\"topology\":[[\"o3\",\"u3\"],[\"u3\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0080","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o4\",\"u4\"],\"topology\":[[\"o4\",\"u4\"],[\"u4\"],[]],\"compact_subset\":[\"u4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o4, u4], [u4], [o4], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o4, u4], [u4], [o4], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123080,"parameters":{"points":["o4","u4"],"topology":[["o4","u4"],["u4"],[]],"compact_subset":["u4"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u4\"],\"points\":[\"o4\",\"u4\"],\"topology\":[[\"o4\",\"u4\"],[\"u4\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0081","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o5\",\"u5\"],\"topology\":[[\"o5\",\"u5\"],[\"u5\"],[]],\"compact_subset\":[\"u5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o5, u5], [u5], [o5], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o5, u5], [u5], [o5], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123081,"parameters":{"points":["o5","u5"],"topology":[["o5","u5"],["u5"],[]],"compact_subset":["u5"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u5\"],\"points\":[\"o5\",\"u5\"],\"topology\":[[\"o5\",\"u5\"],[\"u5\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0082","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o6\",\"u6\"],\"topology\":[[\"o6\",\"u6\"],[\"u6\"],[]],\"compact_subset\":[\"u6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o6, u6], [u6], [o6], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o6, u6], [u6], [o6], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123082,"parameters":{"points":["o6","u6"],"topology":[["o6","u6"],["u6"],[]],"compact_subset":["u6"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u6\"],\"points\":[\"o6\",\"u6\"],\"topology\":[[\"o6\",\"u6\"],[\"u6\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0083","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o7\",\"u7\"],\"topology\":[[\"o7\",\"u7\"],[\"u7\"],[]],\"compact_subset\":[\"u7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o7, u7], [u7], [o7], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o7, u7], [u7], [o7], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123083,"parameters":{"points":["o7","u7"],"topology":[["o7","u7"],["u7"],[]],"compact_subset":["u7"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u7\"],\"points\":[\"o7\",\"u7\"],\"topology\":[[\"o7\",\"u7\"],[\"u7\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0084","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o8\",\"u8\"],\"topology\":[[\"o8\",\"u8\"],[\"u8\"],[]],\"compact_subset\":[\"u8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o8, u8], [u8], [o8], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o8, u8], [u8], [o8], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123084,"parameters":{"points":["o8","u8"],"topology":[["o8","u8"],["u8"],[]],"compact_subset":["u8"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u8\"],\"points\":[\"o8\",\"u8\"],\"topology\":[[\"o8\",\"u8\"],[\"u8\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0085","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o9\",\"u9\"],\"topology\":[[\"o9\",\"u9\"],[\"u9\"],[]],\"compact_subset\":[\"u9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o9, u9], [u9], [o9], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o9, u9], [u9], [o9], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123085,"parameters":{"points":["o9","u9"],"topology":[["o9","u9"],["u9"],[]],"compact_subset":["u9"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u9\"],\"points\":[\"o9\",\"u9\"],\"topology\":[[\"o9\",\"u9\"],[\"u9\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0086","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o10\",\"u10\"],\"topology\":[[\"o10\",\"u10\"],[\"u10\"],[]],\"compact_subset\":[\"u10\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o10, u10], [u10], [o10], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o10, u10], [u10], [o10], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123086,"parameters":{"points":["o10","u10"],"topology":[["o10","u10"],["u10"],[]],"compact_subset":["u10"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u10\"],\"points\":[\"o10\",\"u10\"],\"topology\":[[\"o10\",\"u10\"],[\"u10\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0087","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o11\",\"u11\"],\"topology\":[[\"o11\",\"u11\"],[\"u11\"],[]],\"compact_subset\":[\"u11\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o11, u11], [u11], [o11], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o11, u11], [u11], [o11], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123087,"parameters":{"points":["o11","u11"],"topology":[["o11","u11"],["u11"],[]],"compact_subset":["u11"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u11\"],\"points\":[\"o11\",\"u11\"],\"topology\":[[\"o11\",\"u11\"],[\"u11\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0088","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o12\",\"u12\"],\"topology\":[[\"o12\",\"u12\"],[\"u12\"],[]],\"compact_subset\":[\"u12\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o12, u12], [u12], [o12], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o12, u12], [u12], [o12], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123088,"parameters":{"points":["o12","u12"],"topology":[["o12","u12"],["u12"],[]],"compact_subset":["u12"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u12\"],\"points\":[\"o12\",\"u12\"],\"topology\":[[\"o12\",\"u12\"],[\"u12\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0089","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o13\",\"u13\"],\"topology\":[[\"o13\",\"u13\"],[\"u13\"],[]],\"compact_subset\":[\"u13\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o13, u13], [u13], [o13], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o13, u13], [u13], [o13], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123089,"parameters":{"points":["o13","u13"],"topology":[["o13","u13"],["u13"],[]],"compact_subset":["u13"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u13\"],\"points\":[\"o13\",\"u13\"],\"topology\":[[\"o13\",\"u13\"],[\"u13\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0090","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o14\",\"u14\"],\"topology\":[[\"o14\",\"u14\"],[\"u14\"],[]],\"compact_subset\":[\"u14\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o14, u14], [u14], [o14], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o14, u14], [u14], [o14], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123090,"parameters":{"points":["o14","u14"],"topology":[["o14","u14"],["u14"],[]],"compact_subset":["u14"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u14\"],\"points\":[\"o14\",\"u14\"],\"topology\":[[\"o14\",\"u14\"],[\"u14\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0091","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o15\",\"u15\"],\"topology\":[[\"o15\",\"u15\"],[\"u15\"],[]],\"compact_subset\":[\"u15\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o15, u15], [u15], [o15], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o15, u15], [u15], [o15], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123091,"parameters":{"points":["o15","u15"],"topology":[["o15","u15"],["u15"],[]],"compact_subset":["u15"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u15\"],\"points\":[\"o15\",\"u15\"],\"topology\":[[\"o15\",\"u15\"],[\"u15\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0092","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o16\",\"u16\"],\"topology\":[[\"o16\",\"u16\"],[\"u16\"],[]],\"compact_subset\":[\"u16\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o16, u16], [u16], [o16], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o16, u16], [u16], [o16], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123092,"parameters":{"points":["o16","u16"],"topology":[["o16","u16"],["u16"],[]],"compact_subset":["u16"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u16\"],\"points\":[\"o16\",\"u16\"],\"topology\":[[\"o16\",\"u16\"],[\"u16\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0093","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o17\",\"u17\"],\"topology\":[[\"o17\",\"u17\"],[\"u17\"],[]],\"compact_subset\":[\"u17\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o17, u17], [u17], [o17], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o17, u17], [u17], [o17], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123093,"parameters":{"points":["o17","u17"],"topology":[["o17","u17"],["u17"],[]],"compact_subset":["u17"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u17\"],\"points\":[\"o17\",\"u17\"],\"topology\":[[\"o17\",\"u17\"],[\"u17\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0094","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o18\",\"u18\"],\"topology\":[[\"o18\",\"u18\"],[\"u18\"],[]],\"compact_subset\":[\"u18\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o18, u18], [u18], [o18], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o18, u18], [u18], [o18], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123094,"parameters":{"points":["o18","u18"],"topology":[["o18","u18"],["u18"],[]],"compact_subset":["u18"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u18\"],\"points\":[\"o18\",\"u18\"],\"topology\":[[\"o18\",\"u18\"],[\"u18\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0095","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o19\",\"u19\"],\"topology\":[[\"o19\",\"u19\"],[\"u19\"],[]],\"compact_subset\":[\"u19\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o19, u19], [u19], [o19], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o19, u19], [u19], [o19], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123095,"parameters":{"points":["o19","u19"],"topology":[["o19","u19"],["u19"],[]],"compact_subset":["u19"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u19\"],\"points\":[\"o19\",\"u19\"],\"topology\":[[\"o19\",\"u19\"],[\"u19\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0096","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o20\",\"u20\"],\"topology\":[[\"o20\",\"u20\"],[\"u20\"],[]],\"compact_subset\":[\"u20\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o20, u20], [u20], [o20], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o20, u20], [u20], [o20], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123096,"parameters":{"points":["o20","u20"],"topology":[["o20","u20"],["u20"],[]],"compact_subset":["u20"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u20\"],\"points\":[\"o20\",\"u20\"],\"topology\":[[\"o20\",\"u20\"],[\"u20\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0097","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o21\",\"u21\"],\"topology\":[[\"o21\",\"u21\"],[\"u21\"],[]],\"compact_subset\":[\"u21\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o21, u21], [u21], [o21], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o21, u21], [u21], [o21], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123097,"parameters":{"points":["o21","u21"],"topology":[["o21","u21"],["u21"],[]],"compact_subset":["u21"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u21\"],\"points\":[\"o21\",\"u21\"],\"topology\":[[\"o21\",\"u21\"],[\"u21\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0098","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o22\",\"u22\"],\"topology\":[[\"o22\",\"u22\"],[\"u22\"],[]],\"compact_subset\":[\"u22\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o22, u22], [u22], [o22], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o22, u22], [u22], [o22], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123098,"parameters":{"points":["o22","u22"],"topology":[["o22","u22"],["u22"],[]],"compact_subset":["u22"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u22\"],\"points\":[\"o22\",\"u22\"],\"topology\":[[\"o22\",\"u22\"],[\"u22\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0099","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o23\",\"u23\"],\"topology\":[[\"o23\",\"u23\"],[\"u23\"],[]],\"compact_subset\":[\"u23\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o23, u23], [u23], [o23], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o23, u23], [u23], [o23], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123099,"parameters":{"points":["o23","u23"],"topology":[["o23","u23"],["u23"],[]],"compact_subset":["u23"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u23\"],\"points\":[\"o23\",\"u23\"],\"topology\":[[\"o23\",\"u23\"],[\"u23\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0100","batch_id":"batch-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-subset-closed-in-hausdorff"],"title_zh":"Hausdorff空間中的緊緻集為閉集","stem_zh":"ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={\"points\":[\"o24\",\"u24\"],\"topology\":[[\"o24\",\"u24\"],[\"u24\"],[]],\"compact_subset\":[\"u24\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([o24, u24], [u24], [o24], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)","explanation_zh":"Wave6第六層核對=([o24, u24], [u24], [o24], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)；依完整明示資料獨立重算；對每個x不在K，finite subcover產生與K不交的open neighborhood；empty K亦closed；non-Hausdorff compact subset可非closed，且定理不主張closed implies compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間中的緊緻集為閉集"]},"variant":{"family_id":"qf-mod43-compactclosed-nonhausdorff-boundary","seed":202611123100,"parameters":{"points":["o24","u24"],"topology":[["o24","u24"],["u24"],[]],"compact_subset":["u24"],"domain":"Finite_topological_compact_closed_certificate","operation":"compactclosed-nonhausdorff-boundary","semantic_template":"qf-mod43-compactclosed-nonhausdorff-boundary/{\"compact_subset\":[\"u24\"],\"points\":[\"o24\",\"u24\"],\"topology\":[[\"o24\",\"u24\"],[\"u24\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}