{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0001","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"b\",\"b\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1]], [b, [2, 3, 4, 5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1]], [b, [2, 3, 4, 5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175001,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1]],"y_points":["a","b"],"q_map":["a","a","b","b","b","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"b\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0002","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"b\",\"b\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 2]], [b, [1, 3, 4, 5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 2]], [b, [1, 3, 4, 5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175002,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2]],"y_points":["a","b"],"q_map":["a","b","a","b","b","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0003","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 2]], [b, [3, 4, 5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 2]], [b, [3, 4, 5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175003,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2]],"y_points":["a","b"],"q_map":["a","a","a","b","b","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0004","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,3]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"a\",\"b\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 3]], [b, [1, 2, 4, 5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 3]], [b, [1, 2, 4, 5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175004,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,3]],"y_points":["a","b"],"q_map":["a","b","b","a","b","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,3]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0005","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,3]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"a\",\"b\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 3]], [b, [2, 4, 5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 3]], [b, [2, 4, 5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175005,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,3]],"y_points":["a","b"],"q_map":["a","a","b","a","b","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,3]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0006","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,3]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"a\",\"b\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 2, 3]], [b, [1, 4, 5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 2, 3]], [b, [1, 4, 5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175006,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,3]],"y_points":["a","b"],"q_map":["a","b","a","a","b","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,3]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0007","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,3]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"a\",\"b\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 2, 3]], [b, [4, 5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 2, 3]], [b, [4, 5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175007,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3]],"y_points":["a","b"],"q_map":["a","a","a","a","b","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,3]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0008","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"b\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 4]], [b, [1, 2, 3, 5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 4]], [b, [1, 2, 3, 5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175008,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,4]],"y_points":["a","b"],"q_map":["a","b","b","b","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"b\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0009","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"b\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 4]], [b, [2, 3, 5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 4]], [b, [2, 3, 5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175009,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,4]],"y_points":["a","b"],"q_map":["a","a","b","b","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"b\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0010","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"b\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 2, 4]], [b, [1, 3, 5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 2, 4]], [b, [1, 3, 5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175010,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,4]],"y_points":["a","b"],"q_map":["a","b","a","b","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"b\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0011","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 2, 4]], [b, [3, 5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 2, 4]], [b, [3, 5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175011,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,4]],"y_points":["a","b"],"q_map":["a","a","a","b","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0012","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"a\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 3, 4]], [b, [1, 2, 5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 3, 4]], [b, [1, 2, 5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175012,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,3,4]],"y_points":["a","b"],"q_map":["a","b","b","a","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"a\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,3,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0013","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"a\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 3, 4]], [b, [2, 5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 3, 4]], [b, [2, 5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175013,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,3,4]],"y_points":["a","b"],"q_map":["a","a","b","a","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"a\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,3,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0014","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"a\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 2, 3, 4]], [b, [1, 5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 2, 3, 4]], [b, [1, 5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175014,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,3,4]],"y_points":["a","b"],"q_map":["a","b","a","a","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"a\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,3,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0015","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 2, 3, 4]], [b, [5]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 2, 3, 4]], [b, [5]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175015,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","a","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"a\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,3,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0016","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"b\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 5]], [b, [1, 2, 3, 4]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 5]], [b, [1, 2, 3, 4]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175016,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,5]],"y_points":["a","b"],"q_map":["a","b","b","b","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"b\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0017","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"b\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 5]], [b, [2, 3, 4]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 5]], [b, [2, 3, 4]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175017,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,5]],"y_points":["a","b"],"q_map":["a","a","b","b","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"b\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0018","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"b\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 2, 5]], [b, [1, 3, 4]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 2, 5]], [b, [1, 3, 4]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175018,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,5]],"y_points":["a","b"],"q_map":["a","b","a","b","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0019","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 2, 5]], [b, [3, 4]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 2, 5]], [b, [3, 4]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175019,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,5]],"y_points":["a","b"],"q_map":["a","a","a","b","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0020","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,3,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"a\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 3, 5]], [b, [1, 2, 4]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 3, 5]], [b, [1, 2, 4]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175020,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,3,5]],"y_points":["a","b"],"q_map":["a","b","b","a","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"a\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,3,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0021","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,3,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"a\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 3, 5]], [b, [2, 4]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 3, 5]], [b, [2, 4]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175021,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,3,5]],"y_points":["a","b"],"q_map":["a","a","b","a","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"a\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,3,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0022","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,3,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"a\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 2, 3, 5]], [b, [1, 4]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 2, 3, 5]], [b, [1, 4]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175022,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,3,5]],"y_points":["a","b"],"q_map":["a","b","a","a","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,3,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0023","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,3,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"a\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 2, 3, 5]], [b, [4]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 2, 3, 5]], [b, [4]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175023,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3,5]],"y_points":["a","b"],"q_map":["a","a","a","a","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,3,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0024","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,4,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"b\",\"a\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 4, 5]], [b, [1, 2, 3]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 4, 5]], [b, [1, 2, 3]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175024,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,4,5]],"y_points":["a","b"],"q_map":["a","b","b","b","a","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"b\"],[4,\"a\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,4,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0025","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-fibers；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,4,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"b\",\"a\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 4, 5]], [b, [2, 3]]], 是, 是, fibers_partition_X)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 4, 5]], [b, [2, 3]]], 是, 是, fibers_partition_X)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-fibers","seed":202611175025,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,4,5]],"y_points":["a","b"],"q_map":["a","a","b","b","a","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-fibers","semantic_template":"quotient-fibers|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"b\"],[4,\"a\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,4,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0026","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"b\",\"b\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1]], [b, [2, 3, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1]], [b, [2, 3, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175026,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1]],"y_points":["a","b"],"q_map":["a","a","b","b","b","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"b\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0027","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"b\",\"b\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 2]], [b, [1, 3, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 2]], [b, [1, 3, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175027,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2]],"y_points":["a","b"],"q_map":["a","b","a","b","b","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0028","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 2]], [b, [3, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 2]], [b, [3, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175028,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2]],"y_points":["a","b"],"q_map":["a","a","a","b","b","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0029","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,3]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"a\",\"b\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 3]], [b, [1, 2, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 3]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 3]], [b, [1, 2, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 3]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175029,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,3]],"y_points":["a","b"],"q_map":["a","b","b","a","b","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,3]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0030","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,3]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"a\",\"b\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 3]], [b, [2, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 3]], [b, [2, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175030,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,3]],"y_points":["a","b"],"q_map":["a","a","b","a","b","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,3]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0031","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,3]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"a\",\"b\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 2, 3]], [b, [1, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 3]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 2, 3]], [b, [1, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 3]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175031,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,3]],"y_points":["a","b"],"q_map":["a","b","a","a","b","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,3]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0032","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,3]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"a\",\"b\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 2, 3]], [b, [4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 2, 3]], [b, [4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175032,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3]],"y_points":["a","b"],"q_map":["a","a","a","a","b","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,3]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0033","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"b\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 4]], [b, [1, 2, 3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 4]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 4]], [b, [1, 2, 3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 4]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175033,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,4]],"y_points":["a","b"],"q_map":["a","b","b","b","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"b\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0034","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"b\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 4]], [b, [2, 3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 4]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 4]], [b, [2, 3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 4]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175034,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,4]],"y_points":["a","b"],"q_map":["a","a","b","b","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"b\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0035","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"b\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 2, 4]], [b, [1, 3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 4]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 2, 4]], [b, [1, 3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 4]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175035,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,4]],"y_points":["a","b"],"q_map":["a","b","a","b","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"b\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0036","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 2, 4]], [b, [3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 4]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 2, 4]], [b, [3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 4]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175036,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,4]],"y_points":["a","b"],"q_map":["a","a","a","b","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0037","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"a\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 3, 4]], [b, [1, 2, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 3, 4]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 3, 4]], [b, [1, 2, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 3, 4]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175037,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,3,4]],"y_points":["a","b"],"q_map":["a","b","b","a","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"a\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,3,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0038","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"a\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 3, 4]], [b, [2, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 4]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 3, 4]], [b, [2, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 4]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175038,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,3,4]],"y_points":["a","b"],"q_map":["a","a","b","a","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"a\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,3,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0039","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"a\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 2, 3, 4]], [b, [1, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 4]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 2, 3, 4]], [b, [1, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 4]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175039,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,3,4]],"y_points":["a","b"],"q_map":["a","b","a","a","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"a\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,3,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0040","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 2, 3, 4]], [b, [5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 2, 3, 4]], [b, [5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175040,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","a","a","b"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"a\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,3,4]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0041","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"b\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 5]], [b, [1, 2, 3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 5]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 5]], [b, [1, 2, 3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 5]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175041,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,5]],"y_points":["a","b"],"q_map":["a","b","b","b","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"b\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0042","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"b\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 5]], [b, [2, 3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 5]], [b, [2, 3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175042,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,5]],"y_points":["a","b"],"q_map":["a","a","b","b","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"b\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0043","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"b\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 2, 5]], [b, [1, 3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 5]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 2, 5]], [b, [1, 3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 5]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175043,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,5]],"y_points":["a","b"],"q_map":["a","b","a","b","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0044","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 2, 5]], [b, [3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 2, 5]], [b, [3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175044,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,5]],"y_points":["a","b"],"q_map":["a","a","a","b","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0045","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,3,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"a\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 3, 5]], [b, [1, 2, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 3, 5]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 3, 5]], [b, [1, 2, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 3, 5]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175045,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,3,5]],"y_points":["a","b"],"q_map":["a","b","b","a","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"a\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,3,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0046","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,3,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"a\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 3, 5]], [b, [2, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 3, 5]], [b, [2, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175046,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,3,5]],"y_points":["a","b"],"q_map":["a","a","b","a","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"a\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,3,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0047","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,3,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"a\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 2, 3, 5]], [b, [1, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 5]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 2, 3, 5]], [b, [1, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 5]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175047,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,3,5]],"y_points":["a","b"],"q_map":["a","b","a","a","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,3,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0048","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,3,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"a\",\"b\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 2, 3, 5]], [b, [4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 5]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 2, 3, 5]], [b, [4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 5]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175048,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3,5]],"y_points":["a","b"],"q_map":["a","a","a","a","b","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,3,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0049","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,4,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"b\",\"a\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 4, 5]], [b, [1, 2, 3]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 4, 5]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 4, 5]], [b, [1, 2, 3]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 4, 5]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175049,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,4,5]],"y_points":["a","b"],"q_map":["a","b","b","b","a","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"b\"],[4,\"a\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,4,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0050","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-generation；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,4,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"b\",\"a\",\"a\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[a, [0, 1, 4, 5]], [b, [2, 3]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 4, 5]], 是, 是, U_open_iff_q_preimage_U_open)","explanation_zh":"Wave6第十一層核對=([[a, [0, 1, 4, 5]], [b, [2, 3]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 4, 5]], 是, 是, U_open_iff_q_preimage_U_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-generation","seed":202611175050,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,4,5]],"y_points":["a","b"],"q_map":["a","a","b","b","a","a"],"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-generation","semantic_template":"quotient-topology-generation|{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"b\"],[4,\"a\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,4,5]],y_points:[\"a\",\"b\"]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0051","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"b\",\"b\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175051,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1]],"y_points":["a","b"],"q_map":["a","a","b","b","b","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"b\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0052","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"b\",\"b\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 1], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 1], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175052,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2]],"y_points":["a","b"],"q_map":["a","b","a","b","b","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0053","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 1], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 1], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175053,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2]],"y_points":["a","b"],"q_map":["a","a","a","b","b","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0054","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,3]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"a\",\"b\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175054,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,3]],"y_points":["a","b"],"q_map":["a","b","b","a","b","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,3]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0055","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,3]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"a\",\"b\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175055,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,3]],"y_points":["a","b"],"q_map":["a","a","b","a","b","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,3]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0056","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,3]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"a\",\"b\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175056,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,3]],"y_points":["a","b"],"q_map":["a","b","a","a","b","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,3]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0057","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,3]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"a\",\"b\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175057,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3]],"y_points":["a","b"],"q_map":["a","a","a","a","b","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,3]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0058","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"b\",\"a\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175058,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,4]],"y_points":["a","b"],"q_map":["a","b","b","b","a","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"b\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,4]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0059","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"b\",\"a\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175059,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,4]],"y_points":["a","b"],"q_map":["a","a","b","b","a","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"b\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,4]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0060","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"b\",\"a\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175060,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,4]],"y_points":["a","b"],"q_map":["a","b","a","b","a","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"b\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,4]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0061","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"a\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175061,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,4]],"y_points":["a","b"],"q_map":["a","a","a","b","a","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,4]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0062","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"a\",\"a\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175062,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,3,4]],"y_points":["a","b"],"q_map":["a","b","b","a","a","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"a\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,3,4]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0063","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"a\",\"a\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175063,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,3,4]],"y_points":["a","b"],"q_map":["a","a","b","a","a","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"a\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,3,4]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0064","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"a\",\"a\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175064,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,3,4]],"y_points":["a","b"],"q_map":["a","b","a","a","a","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"a\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,3,4]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0065","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175065,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","a","a","b"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"a\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,3,4]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0066","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"b\",\"b\",\"a\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175066,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,5]],"y_points":["a","b"],"q_map":["a","b","b","b","b","a"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"b\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,5]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0067","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"b\",\"b\",\"a\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175067,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,5]],"y_points":["a","b"],"q_map":["a","a","b","b","b","a"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"b\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,5]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0068","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"b\",\"b\",\"a\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175068,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,5]],"y_points":["a","b"],"q_map":["a","b","a","b","b","a"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,5]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0069","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\",\"a\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175069,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,5]],"y_points":["a","b"],"q_map":["a","a","a","b","b","a"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,5]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0070","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,3,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"a\",\"b\",\"a\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175070,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,3,5]],"y_points":["a","b"],"q_map":["a","b","b","a","b","a"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"a\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,3,5]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0071","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,3,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"a\",\"b\",\"a\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175071,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,3,5]],"y_points":["a","b"],"q_map":["a","a","b","a","b","a"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"a\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,3,5]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0072","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,2,3,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"a\",\"b\",\"a\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175072,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,2,3,5]],"y_points":["a","b"],"q_map":["a","b","a","a","b","a"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,2,3,5]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0073","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,3,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"a\",\"b\",\"a\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175073,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3,5]],"y_points":["a","b"],"q_map":["a","a","a","a","b","a"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,3,5]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0074","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,4,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"b\",\"a\",\"a\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 1], [4, 0], [5, 0]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 1], [4, 0], [5, 0]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175074,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,4,5]],"y_points":["a","b"],"q_map":["a","b","b","b","a","a"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"b\"],[4,\"a\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,4,5]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0075","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-final-continuity；資料={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,4,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"b\",\"a\",\"a\"],\"z_points\":[0,1],\"z_topology\":[[],[1],[0,1]],\"h_map\":[0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 0], [5, 0]], quotient_final_property_for_surjective_q)","explanation_zh":"Wave6第十一層核對=(否, 否, 是, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 0], [5, 0]], quotient_final_property_for_surjective_q)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-final-continuity","seed":202611175075,"parameters":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,4,5]],"y_points":["a","b"],"q_map":["a","a","b","b","a","a"],"z_points":[0,1],"z_topology":[[],[1],[0,1]],"h_map":[0,1],"domain":"Complete_finite_quotient_topology","operation":"quotient-final-continuity","semantic_template":"quotient-final-continuity|{h_map:[[\"a\",0],[\"b\",1]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"b\"],[4,\"a\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,4,5]],y_points:[\"a\",\"b\"],z_points:[0,1],z_topology:[[],[0,1],[1]]}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0076","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"non_surjective\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0],[0,1,2,3]],\"y_points\":[\"y0\",\"y1\",\"y2\",\"y3\",\"y4\"],\"q_map\":[\"y0\",\"y1\",\"y2\",\"y3\"]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y4], [y0], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)","explanation_zh":"Wave6第十一層核對=(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y4], [y0], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175076,"parameters":{"mode":"non_surjective","model":{"x_points":[0,1,2,3],"x_topology":[[],[0],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"non_surjective\",model:{q_map:[[0,\"y0\"],[1,\"y1\"],[2,\"y2\"],[3,\"y3\"]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0]],y_points:[\"y0\",\"y1\",\"y2\",\"y3\",\"y4\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0077","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"open_image_error\",\"model\":{\"x_points\":[0,1,2,3,4],\"x_topology\":[[],[0],[0,1,2,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\"],\"subset\":[0]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(open_image_error, [0], 是, [a], [0, 1, 2], 否, 否, quotient_maps_need_not_be_open)","explanation_zh":"Wave6第十一層核對=(open_image_error, [0], 是, [a], [0, 1, 2], 否, 否, quotient_maps_need_not_be_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175077,"parameters":{"mode":"open_image_error","model":{"x_points":[0,1,2,3,4],"x_topology":[[],[0],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","b","b"],"subset":[0]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"open_image_error\",model:{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"]],subset:[0],x_points:[0,1,2,3,4],x_topology:[[],[0,1,2,3,4],[0]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0078","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"coarser_target\",\"model\":{\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\",\"b\"],\"candidate_topology\":[[],[\"a\",\"b\"]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(coarser_target, [[a, b], []], [[a, b], [a], []], 是, 是, 否, continuity_alone_does_not_identify_the_final_topology)","explanation_zh":"Wave6第十一層核對=(coarser_target, [[a, b], []], [[a, b], [a], []], 是, 是, 否, continuity_alone_does_not_identify_the_final_topology)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175078,"parameters":{"mode":"coarser_target","model":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2]],"y_points":["a","b"],"q_map":["a","a","a","b","b","b"],"candidate_topology":[[],["a","b"]]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"coarser_target\",model:{candidate_topology:[[\"a\",\"b\"],[]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0079","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"fiber_split\",\"model\":{\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"b\",\"b\",\"b\"],\"subset\":[0]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(fiber_split, [0], [a], [0, 1], 否, 否, only_saturated_subsets_are_full_preimages)","explanation_zh":"Wave6第十一層核對=(fiber_split, [0], [a], [0, 1], 否, 否, only_saturated_subsets_are_full_preimages)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175079,"parameters":{"mode":"fiber_split","model":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5]],"y_points":["a","b"],"q_map":["a","a","b","b","b","b"],"subset":[0]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"fiber_split\",model:{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"b\"],[4,\"b\"],[5,\"b\"]],subset:[0],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0080","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"non_surjective\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[1],[0,1,2,3]],\"y_points\":[\"y0\",\"y1\",\"y2\",\"y3\",\"y4\"],\"q_map\":[\"y0\",\"y1\",\"y2\",\"y3\"]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y1, y4], [y1], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)","explanation_zh":"Wave6第十一層核對=(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y1, y4], [y1], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175080,"parameters":{"mode":"non_surjective","model":{"x_points":[0,1,2,3],"x_topology":[[],[1],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"non_surjective\",model:{q_map:[[0,\"y0\"],[1,\"y1\"],[2,\"y2\"],[3,\"y3\"]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1]],y_points:[\"y0\",\"y1\",\"y2\",\"y3\",\"y4\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0081","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"open_image_error\",\"model\":{\"x_points\":[0,1,2,3,4],\"x_topology\":[[],[1],[0,1,2,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\"],\"subset\":[1]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(open_image_error, [1], 是, [a], [0, 1, 2], 否, 否, quotient_maps_need_not_be_open)","explanation_zh":"Wave6第十一層核對=(open_image_error, [1], 是, [a], [0, 1, 2], 否, 否, quotient_maps_need_not_be_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175081,"parameters":{"mode":"open_image_error","model":{"x_points":[0,1,2,3,4],"x_topology":[[],[1],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","b","b"],"subset":[1]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"open_image_error\",model:{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"]],subset:[1],x_points:[0,1,2,3,4],x_topology:[[],[0,1,2,3,4],[1]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0082","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"coarser_target\",\"model\":{\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,3]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"a\",\"b\",\"b\"],\"candidate_topology\":[[],[\"a\",\"b\"]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(coarser_target, [[a, b], []], [[a, b], [a], []], 是, 是, 否, continuity_alone_does_not_identify_the_final_topology)","explanation_zh":"Wave6第十一層核對=(coarser_target, [[a, b], []], [[a, b], [a], []], 是, 是, 否, continuity_alone_does_not_identify_the_final_topology)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175082,"parameters":{"mode":"coarser_target","model":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3]],"y_points":["a","b"],"q_map":["a","a","a","a","b","b"],"candidate_topology":[[],["a","b"]]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"coarser_target\",model:{candidate_topology:[[\"a\",\"b\"],[]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,3]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0083","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"fiber_split\",\"model\":{\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"b\",\"b\",\"b\"],\"subset\":[0]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(fiber_split, [0], [a], [0, 2], 否, 否, only_saturated_subsets_are_full_preimages)","explanation_zh":"Wave6第十一層核對=(fiber_split, [0], [a], [0, 2], 否, 否, only_saturated_subsets_are_full_preimages)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175083,"parameters":{"mode":"fiber_split","model":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5]],"y_points":["a","b"],"q_map":["a","b","a","b","b","b"],"subset":[0]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"fiber_split\",model:{q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"b\"]],subset:[0],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0084","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"non_surjective\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1],[0,1,2,3]],\"y_points\":[\"y0\",\"y1\",\"y2\",\"y3\",\"y4\"],\"q_map\":[\"y0\",\"y1\",\"y2\",\"y3\"]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y1, y4], [y0, y1], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)","explanation_zh":"Wave6第十一層核對=(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y1, y4], [y0, y1], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175084,"parameters":{"mode":"non_surjective","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"non_surjective\",model:{q_map:[[0,\"y0\"],[1,\"y1\"],[2,\"y2\"],[3,\"y3\"]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1]],y_points:[\"y0\",\"y1\",\"y2\",\"y3\",\"y4\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0085","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"open_image_error\",\"model\":{\"x_points\":[0,1,2,3,4],\"x_topology\":[[],[0,1],[0,1,2,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\"],\"subset\":[0,1]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(open_image_error, [0, 1], 是, [a], [0, 1, 2], 否, 否, quotient_maps_need_not_be_open)","explanation_zh":"Wave6第十一層核對=(open_image_error, [0, 1], 是, [a], [0, 1, 2], 否, 否, quotient_maps_need_not_be_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175085,"parameters":{"mode":"open_image_error","model":{"x_points":[0,1,2,3,4],"x_topology":[[],[0,1],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","b","b"],"subset":[0,1]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"open_image_error\",model:{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"]],subset:[0,1],x_points:[0,1,2,3,4],x_topology:[[],[0,1,2,3,4],[0,1]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0086","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"coarser_target\",\"model\":{\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"a\",\"b\"],\"candidate_topology\":[[],[\"a\",\"b\"]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(coarser_target, [[a, b], []], [[a, b], [a], []], 是, 是, 否, continuity_alone_does_not_identify_the_final_topology)","explanation_zh":"Wave6第十一層核對=(coarser_target, [[a, b], []], [[a, b], [a], []], 是, 是, 否, continuity_alone_does_not_identify_the_final_topology)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175086,"parameters":{"mode":"coarser_target","model":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,4]],"y_points":["a","b"],"q_map":["a","a","a","b","a","b"],"candidate_topology":[[],["a","b"]]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"coarser_target\",model:{candidate_topology:[[\"a\",\"b\"],[]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,4]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0087","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"fiber_split\",\"model\":{\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\",\"b\"],\"subset\":[0]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(fiber_split, [0], [a], [0, 1, 2], 否, 否, only_saturated_subsets_are_full_preimages)","explanation_zh":"Wave6第十一層核對=(fiber_split, [0], [a], [0, 1, 2], 否, 否, only_saturated_subsets_are_full_preimages)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175087,"parameters":{"mode":"fiber_split","model":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5]],"y_points":["a","b"],"q_map":["a","a","a","b","b","b"],"subset":[0]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"fiber_split\",model:{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"b\"]],subset:[0],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0088","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"non_surjective\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[2],[0,1,2,3]],\"y_points\":[\"y0\",\"y1\",\"y2\",\"y3\",\"y4\"],\"q_map\":[\"y0\",\"y1\",\"y2\",\"y3\"]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y2, y4], [y2], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)","explanation_zh":"Wave6第十一層核對=(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y2, y4], [y2], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175088,"parameters":{"mode":"non_surjective","model":{"x_points":[0,1,2,3],"x_topology":[[],[2],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"non_surjective\",model:{q_map:[[0,\"y0\"],[1,\"y1\"],[2,\"y2\"],[3,\"y3\"]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[2]],y_points:[\"y0\",\"y1\",\"y2\",\"y3\",\"y4\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0089","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"open_image_error\",\"model\":{\"x_points\":[0,1,2,3,4],\"x_topology\":[[],[2],[0,1,2,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\"],\"subset\":[2]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(open_image_error, [2], 是, [a], [0, 1, 2], 否, 否, quotient_maps_need_not_be_open)","explanation_zh":"Wave6第十一層核對=(open_image_error, [2], 是, [a], [0, 1, 2], 否, 否, quotient_maps_need_not_be_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175089,"parameters":{"mode":"open_image_error","model":{"x_points":[0,1,2,3,4],"x_topology":[[],[2],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","b","b"],"subset":[2]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"open_image_error\",model:{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"]],subset:[2],x_points:[0,1,2,3,4],x_topology:[[],[0,1,2,3,4],[2]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0090","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"coarser_target\",\"model\":{\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],\"candidate_topology\":[[],[\"a\",\"b\"]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(coarser_target, [[a, b], []], [[a, b], [a], []], 是, 是, 否, continuity_alone_does_not_identify_the_final_topology)","explanation_zh":"Wave6第十一層核對=(coarser_target, [[a, b], []], [[a, b], [a], []], 是, 是, 否, continuity_alone_does_not_identify_the_final_topology)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175090,"parameters":{"mode":"coarser_target","model":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","a","a","b"],"candidate_topology":[[],["a","b"]]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"coarser_target\",model:{candidate_topology:[[\"a\",\"b\"],[]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"a\"],[4,\"a\"],[5,\"b\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,3,4]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0091","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"fiber_split\",\"model\":{\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"b\",\"a\",\"b\",\"b\"],\"subset\":[0]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(fiber_split, [0], [a], [0, 3], 否, 否, only_saturated_subsets_are_full_preimages)","explanation_zh":"Wave6第十一層核對=(fiber_split, [0], [a], [0, 3], 否, 否, only_saturated_subsets_are_full_preimages)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175091,"parameters":{"mode":"fiber_split","model":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5]],"y_points":["a","b"],"q_map":["a","b","b","a","b","b"],"subset":[0]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"fiber_split\",model:{q_map:[[0,\"a\"],[1,\"b\"],[2,\"b\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],subset:[0],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0092","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"non_surjective\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,2],[0,1,2,3]],\"y_points\":[\"y0\",\"y1\",\"y2\",\"y3\",\"y4\"],\"q_map\":[\"y0\",\"y1\",\"y2\",\"y3\"]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y2, y4], [y0, y2], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)","explanation_zh":"Wave6第十一層核對=(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y2, y4], [y0, y2], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175092,"parameters":{"mode":"non_surjective","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,2],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"non_surjective\",model:{q_map:[[0,\"y0\"],[1,\"y1\"],[2,\"y2\"],[3,\"y3\"]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2]],y_points:[\"y0\",\"y1\",\"y2\",\"y3\",\"y4\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0093","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"open_image_error\",\"model\":{\"x_points\":[0,1,2,3,4],\"x_topology\":[[],[0,2],[0,1,2,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\"],\"subset\":[0,2]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(open_image_error, [0, 2], 是, [a], [0, 1, 2], 否, 否, quotient_maps_need_not_be_open)","explanation_zh":"Wave6第十一層核對=(open_image_error, [0, 2], 是, [a], [0, 1, 2], 否, 否, quotient_maps_need_not_be_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175093,"parameters":{"mode":"open_image_error","model":{"x_points":[0,1,2,3,4],"x_topology":[[],[0,2],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","b","b"],"subset":[0,2]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"open_image_error\",model:{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"]],subset:[0,2],x_points:[0,1,2,3,4],x_topology:[[],[0,1,2,3,4],[0,2]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0094","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"coarser_target\",\"model\":{\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\",\"a\"],\"candidate_topology\":[[],[\"a\",\"b\"]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(coarser_target, [[a, b], []], [[a, b], [a], []], 是, 是, 否, continuity_alone_does_not_identify_the_final_topology)","explanation_zh":"Wave6第十一層核對=(coarser_target, [[a, b], []], [[a, b], [a], []], 是, 是, 否, continuity_alone_does_not_identify_the_final_topology)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175094,"parameters":{"mode":"coarser_target","model":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,5]],"y_points":["a","b"],"q_map":["a","a","a","b","b","a"],"candidate_topology":[[],["a","b"]]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"coarser_target\",model:{candidate_topology:[[\"a\",\"b\"],[]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,5]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0095","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"fiber_split\",\"model\":{\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"b\",\"a\",\"b\",\"b\"],\"subset\":[0]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(fiber_split, [0], [a], [0, 1, 3], 否, 否, only_saturated_subsets_are_full_preimages)","explanation_zh":"Wave6第十一層核對=(fiber_split, [0], [a], [0, 1, 3], 否, 否, only_saturated_subsets_are_full_preimages)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175095,"parameters":{"mode":"fiber_split","model":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5]],"y_points":["a","b"],"q_map":["a","a","b","a","b","b"],"subset":[0]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"fiber_split\",model:{q_map:[[0,\"a\"],[1,\"a\"],[2,\"b\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],subset:[0],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0096","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"non_surjective\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[1,2],[0,1,2,3]],\"y_points\":[\"y0\",\"y1\",\"y2\",\"y3\",\"y4\"],\"q_map\":[\"y0\",\"y1\",\"y2\",\"y3\"]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y1, y2, y4], [y1, y2], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)","explanation_zh":"Wave6第十一層核對=(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y1, y2, y4], [y1, y2], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175096,"parameters":{"mode":"non_surjective","model":{"x_points":[0,1,2,3],"x_topology":[[],[1,2],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"non_surjective\",model:{q_map:[[0,\"y0\"],[1,\"y1\"],[2,\"y2\"],[3,\"y3\"]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2]],y_points:[\"y0\",\"y1\",\"y2\",\"y3\",\"y4\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0097","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"open_image_error\",\"model\":{\"x_points\":[0,1,2,3,4],\"x_topology\":[[],[1,2],[0,1,2,3,4]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"b\",\"b\"],\"subset\":[1,2]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(open_image_error, [1, 2], 是, [a], [0, 1, 2], 否, 否, quotient_maps_need_not_be_open)","explanation_zh":"Wave6第十一層核對=(open_image_error, [1, 2], 是, [a], [0, 1, 2], 否, 否, quotient_maps_need_not_be_open)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175097,"parameters":{"mode":"open_image_error","model":{"x_points":[0,1,2,3,4],"x_topology":[[],[1,2],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","b","b"],"subset":[1,2]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"open_image_error\",model:{q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"b\"],[4,\"b\"]],subset:[1,2],x_points:[0,1,2,3,4],x_topology:[[],[0,1,2,3,4],[1,2]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0098","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"coarser_target\",\"model\":{\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5],[0,1,2,3,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"a\",\"a\",\"a\",\"b\",\"a\"],\"candidate_topology\":[[],[\"a\",\"b\"]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(coarser_target, [[a, b], []], [[a, b], [a], []], 是, 是, 否, continuity_alone_does_not_identify_the_final_topology)","explanation_zh":"Wave6第十一層核對=(coarser_target, [[a, b], []], [[a, b], [a], []], 是, 是, 否, continuity_alone_does_not_identify_the_final_topology)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175098,"parameters":{"mode":"coarser_target","model":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3,5]],"y_points":["a","b"],"q_map":["a","a","a","a","b","a"],"candidate_topology":[[],["a","b"]]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"coarser_target\",model:{candidate_topology:[[\"a\",\"b\"],[]],q_map:[[0,\"a\"],[1,\"a\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"a\"]],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5],[0,1,2,3,5]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0099","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"fiber_split\",\"model\":{\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[],[0,1,2,3,4,5]],\"y_points\":[\"a\",\"b\"],\"q_map\":[\"a\",\"b\",\"a\",\"a\",\"b\",\"b\"],\"subset\":[0]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(fiber_split, [0], [a], [0, 2, 3], 否, 否, only_saturated_subsets_are_full_preimages)","explanation_zh":"Wave6第十一層核對=(fiber_split, [0], [a], [0, 2, 3], 否, 否, only_saturated_subsets_are_full_preimages)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175099,"parameters":{"mode":"fiber_split","model":{"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5]],"y_points":["a","b"],"q_map":["a","b","a","a","b","b"],"subset":[0]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"fiber_split\",model:{q_map:[[0,\"a\"],[1,\"b\"],[2,\"a\"],[3,\"a\"],[4,\"b\"],[5,\"b\"]],subset:[0],x_points:[0,1,2,3,4,5],x_topology:[[],[0,1,2,3,4,5]],y_points:[\"a\",\"b\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r05-quotient-topology-0100","batch_id":"batch-20261015-advance-20261117-r05-quotient-topology-foundations","language":"zh-Hant","source_mko_ids":["mko-quotient-topology"],"title_zh":"商拓撲","stem_zh":"finite total surjection、fibers與U open iff q-preimage(U) open責任。操作=quotient-topology-boundary；資料={\"mode\":\"non_surjective\",\"model\":{\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2],[0,1,2,3]],\"y_points\":[\"y0\",\"y1\",\"y2\",\"y3\",\"y4\"],\"q_map\":[\"y0\",\"y1\",\"y2\",\"y3\"]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y1, y2, y4], [y0, y1, y2], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)","explanation_zh":"Wave6第十一層核對=(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], 否, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y1, y2, y4], [y0, y1, y2], [y4], []], 是, 否, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)；依完整明示資料獨立重算；只由saturated opens對應quotient opens，final continuity需完整preimage audit；open images、非滿射或任意coarser topology不等於quotient topology。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["商拓撲"]},"variant":{"family_id":"qf-mod48-quotient-topology-boundary","seed":202611175100,"parameters":{"mode":"non_surjective","model":{"x_points":[0,1,2,3],"x_topology":[[],[0,1,2],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]},"domain":"Complete_finite_quotient_topology","operation":"quotient-topology-boundary","semantic_template":"quotient-topology-boundary|{mode:\"non_surjective\",model:{q_map:[[0,\"y0\"],[1,\"y1\"],[2,\"y2\"],[3,\"y3\"]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2]],y_points:[\"y0\",\"y1\",\"y2\",\"y3\",\"y4\"]}}"}},"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；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}