{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0001","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025001,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0002","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025002,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0003","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025003,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0004","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025004,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0005","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025005,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0006","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025006,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0007","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025007,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0008","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025008,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0009","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025009,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0010","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025010,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0011","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025011,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0012","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025012,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0013","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025013,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0014","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025014,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0015","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025015,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0016","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025016,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0017","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025017,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0018","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025018,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0019","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025019,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0020","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025020,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0021","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025021,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0022","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025022,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0023","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025023,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0024","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025024,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0025","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-finite-space-certificate；資料={\"n\":5,\"topology\":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]]}。請依 [carrier_size,open_set_count,compact,finite_topology_certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)","explanation_zh":"Wave5第二層核對=(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-finite-space-certificate","seed":202611025025,"parameters":{"n":5,"topology":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-finite-space-certificate","semantic_template":"qf-mod33-compact-finite-space-certificate/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[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":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0026","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025026,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0027","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025027,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,1],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0028","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025028,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0029","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025029,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,1],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0030","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025030,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,1],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0031","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025031,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,1],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0032","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025032,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,2],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0033","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025033,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,2],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0034","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025034,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,2],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0035","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025035,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,2],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0036","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025036,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,2],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0037","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025037,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,2],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0038","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025038,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,3],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0039","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025039,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,3],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0040","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025040,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,3],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0041","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025041,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,3],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0042","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025042,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,3],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0043","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025043,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,3],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0044","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025044,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,4],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0045","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025045,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,4],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0046","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025046,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,4],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0047","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025047,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0048","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025048,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,4],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0049","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025049,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0050","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-cover-audit；資料={\"n\":5,\"topology\":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5第二層核對=([], [0, 1, 2, 3, 4], 是, 是)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-cover-audit","seed":202611025050,"parameters":{"n":5,"topology":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_compactness","operation":"compact-cover-audit","semantic_template":"qf-mod33-compact-cover-audit/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[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":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0051","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 2, 3], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 2, 3], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025051,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0052","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 2, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 2, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025052,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,1],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0053","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 2, 3], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 2, 3], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025053,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0054","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 3, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 3, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025054,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,1],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0055","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 2, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 2, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025055,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,1],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0056","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 3, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 3, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025056,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,1],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0057","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 2, 3], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 2, 3], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025057,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,2],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0058","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 2, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 2, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025058,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,2],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0059","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 2, 3], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 2, 3], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025059,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,2],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0060","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 2, 3, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 2, 3, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025060,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,2],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0061","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 2, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 2, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025061,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,2],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0062","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 2, 3, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 2, 3, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025062,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,2],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0063","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 2, 3], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 2, 3], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025063,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,3],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0064","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 3, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 3, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025064,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,3],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0065","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 2, 3], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 2, 3], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025065,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,3],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0066","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 2, 3, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 2, 3, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025066,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,3],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0067","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 3, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 3, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025067,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,3],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0068","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 2, 3, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 2, 3, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025068,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,3],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0069","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 2, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 2, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025069,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,4],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0070","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 3, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 3, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025070,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,4],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0071","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 2, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 2, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025071,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,4],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0072","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 2, 3, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 2, 3, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025072,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0073","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 3, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 3, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025073,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,4],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0074","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 2, 3, 4], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 2, 3, 4], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025074,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0075","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-subcover-witness；資料={\"n\":5,\"topology\":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,covered_union,valid_subcover,scope] 欄序回答。","answer":{"kind":"expression","value":"([1], [0, 1, 2, 3], 是, finite_subcover_for_supplied_cover)","explanation_zh":"Wave5第二層核對=([1], [0, 1, 2, 3], 是, finite_subcover_for_supplied_cover)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-subcover-witness","seed":202611025075,"parameters":{"n":5,"topology":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-subcover-witness","semantic_template":"qf-mod33-compact-subcover-witness/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[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":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0076","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025076,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0077","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025077,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,1],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0078","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025078,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0079","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025079,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,1],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0080","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025080,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,1],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0081","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025081,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,1],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0082","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025082,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,2],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0083","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025083,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,2],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0084","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025084,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,2],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0085","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025085,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,2],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0086","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025086,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,2],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0087","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025087,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,2],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0088","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025088,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,3],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0089","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025089,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,3],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0090","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025090,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,3],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0091","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025091,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,3],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0092","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025092,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,3],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0093","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025093,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,3],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0094","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025094,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,4],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0095","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025095,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,4],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0096","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025096,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,4],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0097","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025097,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0098","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025098,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,4],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0099","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025099,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r05-compact-topological-space-0100","batch_id":"batch-20261014-advance-20261102-r05-compact-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-compact-topological-space"],"title_zh":"緊緻拓撲空間","stem_zh":"完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)","explanation_zh":"Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻拓撲空間"]},"variant":{"family_id":"qf-mod33-compact-single-cover-boundary","seed":202611025100,"parameters":{"n":5,"topology":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topology_compactness","operation":"compact-single-cover-boundary","semantic_template":"qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[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":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}