{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261102-r05-compact-topological-space-0008", "pilot_number": "R0032-16833", "number": 16833, "family_id": "qf-mod33-compact-finite-space-certificate", "title": { "zh-Hant": "第 16833 題：緊緻拓撲空間", "en": "Question 16833: Finiteness Supplies the Universal Compactness Certificate" }, "original_question": { "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" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261102-r05-compact-topological-space-0008", "batch_id": "batch-20261014-advance-20261102-r05-compact-topological-space-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "4c74ee5cb44c4df2a64e8ed6e8a2c3b41448ee879cd3f85bca7f9fcb5708f557", "family_id": "qf-mod33-compact-finite-space-certificate", "seed": 202611025008, "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]]}" }, "fields": { "title": "Finiteness Supplies the Universal Compactness Certificate", "stem": "Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Give (carrier_size,open_set_count,compact?,every_open_cover_is_a_subfamily_of_a_finite_topology).", "answer": { "kind": "expression", "value": "(5, 6, Yes, every_open_cover_is_a_subfamily_of_a_finite_topology)", "explanation": "The complete topology has finitely many distinct open sets. Every set-family open cover is therefore finite; an indexed cover with repetitions can select representatives of its finitely many distinct members. This finite structural argument covers every open cover, unlike a single tested cover. It is not a new Lean receipt or an infinite-space compactness certificate. The exact result, retaining the requested order, is (5, 6, Yes, every_open_cover_is_a_subfamily_of_a_finite_topology)." }, "skills": [ "Finiteness Supplies the Universal Compactness Certificate" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "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。", "en": "Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Give (carrier_size,open_set_count,compact?,every_open_cover_is_a_subfamily_of_a_finite_topology)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "same complete carrier Fin(5);original ordered complete topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]", "after": "unique empty/whole opens=是;ordered open pairs checked=36;first union failure=null;first intersection failure=null", "relation": "implies", "rule_id": "topology_complete_finite_parent_all_pairs", "conditions": [], "explanation": { "zh-Hant": "個別子集須照原載體遞增且無重複，拓撲族須有唯一空集、全集並核全部二元聯／交。因不同開集有限，任意聯只涉及有限種開集，二元封閉加空集便足夠；不另供度量。保留拓撲原順序。", "en": "Each subset is increasing and duplicate-free.The topology has unique empty/whole opens and closure under every binary union/intersection.Finitely many distinct opens reduce arbitrary unions to finite ones,handled by binary closure and the empty set.No metric is supplied.Keep original topology order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "complete finite topology contains exactly6 distinct opens on Fin(5)", "after": "for ANY set-family open cover U,each member lies in this finite topology;U is finite and covers by itself", "relation": "implies", "rule_id": "compact_every_open_cover_finite_topology_not_test_suite", "conditions": [], "explanation": { "zh-Hant": "此處量化任意開覆蓋，不是只枚舉幾份測試；每個成員在已列有限拓撲中，故不同成員至多開集總數，這份覆蓋本身已有有限子覆蓋。", "en": "Quantify over any open cover,not a test suite.Every member belongs to the supplied finite topology,so the set of distinct members is finite and already constitutes a finite subcover." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "same finite topology with6 distinct opens;an arbitrary INDEXED cover may repeat opens", "after": "select one original index per distinct open used;only finitely many choices;the representative family retains the same union", "relation": "implies", "rule_id": "compact_indexed_duplicate_cover_finite_representatives", "conditions": [], "explanation": { "zh-Hant": "帶索引覆蓋即使無限重複，僅有限種不同開集；每種選一個原索引就保同聯集且有限。不把索引數可能無限誤當不同開集無限，也不外推一般無限空間。", "en": "An indexed cover may repeat opens infinitely,but only finitely many distinct opens occur.One original index per distinct open retains the union and gives a finite subcover.Do not confuse infinitely many indices with infinitely many distinct opens or extend this argument to arbitrary infinite spaces." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ 5, 6, true, "every_open_cover_is_a_subfamily_of_a_finite_topology" ] }, "conclusion": { "canonical_value": "(5, 6, 是, every_open_cover_is_a_subfamily_of_a_finite_topology)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261102-r05-compact-topological-space-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261102-r05-compact-topological-space-foundations/questions.jsonl", "batch_sha256": "01b7b492658c47414f5f927eb995ff63a21e4da461bafc0cf2fb9f40397a1db2", "original_line": 8, "original_line_sha256": "36a9f920e1e6bfc21c7c52577a4c5383716765a0a6f9c825f43d13c3cff90b3c", "question_compact_sha256": "4c74ee5cb44c4df2a64e8ed6e8a2c3b41448ee879cd3f85bca7f9fcb5708f557", "english_row_compact_sha256": "407613370a8fa1c308d57d46163a7f52f12ce0a598477bded17442d4eb56a543", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261102-r05-compact-topological-space-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261102-r05-compact-topological-space-foundations.en.jsonl" }, "mko_bindings": { "mko-compact-topological-space": { "sha256": "13f302a171c38a633a5a145da984410cab9bef79a09c4aef91bdaf9e016226e6", "english_sha256": "cb6c5eefd1b67e492a6fcd011cf21c954ed3dcfdf9a15d5bbc46db1460c8e204", "original_path": "public/data/mko/compact-topological-space.json", "english_path": "public/data/i18n/editions/mko-compact-topological-space.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "fe4c48735753bdd58c10db9b2ca3a2a3baeb31d0424c3fc09dea62317185ddae", "original_candidate_sha256": "ec9b44dd510577b9f55e6d6c4b3606d7263379015d5993b6273372514808afc5", "original_candidate_line_sha256": "ec9b44dd510577b9f55e6d6c4b3606d7263379015d5993b6273372514808afc5" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-007-p33", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e", "question_first_witness_sha256": "790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "a09c1a2450b60d0f69aa555234750e0643daa8c45e9aafef11794385b0e869aa", "arithmetic_steps": 0 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }