{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261111-r03-hausdorff-space-0046", "pilot_number": "R0044-22921", "number": 22921, "family_id": "qf-mod42-hausdorff-full-audit", "title": { "zh-Hant": "第 22921 題：Hausdorff空間", "en": "Question 22921: All Distinct Point Pairs Must Be Separated in the Complete Finite Topology" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261111-r03-hausdorff-space-0046", "batch_id": "batch-20261014-advance-20261111-r03-hausdorff-space-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-hausdorff-space" ], "title_zh": "Hausdorff空間", "stem_zh": "完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([a20, b20, c20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], 是, all_distinct_pairs_exhaustively_separated)", "explanation_zh": "Wave6第五層核對=([a20, b20, c20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "Hausdorff空間" ] }, "variant": { "family_id": "qf-mod42-hausdorff-full-audit", "seed": 202611113046, "parameters": { "points": [ "a20", "b20", "c20" ], "topology": [ [ "a20", "b20", "c20" ], [ "a20", "b20" ], [ "a20", "c20" ], [ "a20" ], [ "b20", "c20" ], [ "b20" ], [ "c20" ], [] ], "domain": "Complete_finite_topology_Hausdorff_audit", "operation": "hausdorff-full-audit", "semantic_template": "qf-mod42-hausdorff-full-audit/{\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或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-20261111-r03-hausdorff-space-0046", "batch_id": "batch-20261014-advance-20261111-r03-hausdorff-space-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "3c628409a9a487bd476cea557a3bcf9a2d4dcfddc49badf15dd7bcc102b95abb", "family_id": "qf-mod42-hausdorff-full-audit", "seed": 202611113046, "semantic_template": "qf-mod42-hausdorff-full-audit/{\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]]}" }, "fields": { "title": "All Distinct Point Pairs Must Be Separated in the Complete Finite Topology", "stem": "Use finite unique labeled points=[a20, b20, c20] and the complete canonical topology=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).", "answer": { "kind": "expression", "value": "([a20, b20, c20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], Yes, all_distinct_pairs_exhaustively_separated)", "explanation": "Enumerate every unordered pair in original point order and its first source-order witness. A single failed pair disproves Hausdorffness. The empty/singleton cases have no distinct pair and are vacuously Hausdorff. Finite Hausdorff implies discrete, but that corollary must not be extrapolated to infinite spaces. The exact result, retaining the requested order, is ([a20, b20, c20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], Yes, all_distinct_pairs_exhaustively_separated)." }, "skills": [ "All Distinct Point Pairs Must Be Separated in the Complete Finite Topology" ], "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": "完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use finite unique labeled points=[a20, b20, c20] and the complete canonical topology=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=hausdorff-full-audit;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[points, pair_witness_rows, Hausdorff, certificate]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "保留原題歷史介面標記，直接列來源欄序，不需JavaScript。tag不是額外證據，固定No依原聲稱範圍解讀；排序輔助只處理協定，不是數學oracle。", "en": "Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness." }, "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": "ORIGINAL carrier order=[a20, b20, c20];source canonical open family=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []];actual open index=0,U=[a20, b20, c20]", "after": "ALL [other_index,union,intersection] rows=[[0, [a20, b20, c20], [a20, b20, c20]], [1, [a20, b20, c20], [a20, b20]], [2, [a20, b20, c20], [a20, c20]], [3, [a20, b20, c20], [a20]], [4, [a20, b20, c20], [b20, c20]], [5, [a20, b20, c20], [b20]], [6, [a20, b20, c20], [c20]], [7, [a20, b20, c20], []]];empty/full present;family qualification passed with exact source English collation", "relation": "implies", "rule_id": "hausdorff_complete_labeled_topology_and_source_order", "conditions": [], "explanation": { "zh-Hant": "點標籤與原carrier次序保留；各open內依來源字串排序，family依固定English collation，不按大小或另一個常見慣例。逐全部open pair核公理，空carrier也有拓撲[[]]。", "en": "Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]." }, "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": "ORIGINAL carrier order=[a20, b20, c20];source canonical open family=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []];actual open index=1,U=[a20, b20]", "after": "ALL [other_index,union,intersection] rows=[[0, [a20, b20, c20], [a20, b20]], [1, [a20, b20], [a20, b20]], [2, [a20, b20, c20], [a20]], [3, [a20, b20], [a20]], [4, [a20, b20, c20], [b20]], [5, [a20, b20], [b20]], [6, [a20, b20, c20], []], [7, [a20, b20], []]];empty/full present;family qualification passed with exact source English collation", "relation": "implies", "rule_id": "hausdorff_complete_labeled_topology_and_source_order", "conditions": [], "explanation": { "zh-Hant": "點標籤與原carrier次序保留；各open內依來源字串排序，family依固定English collation，不按大小或另一個常見慣例。逐全部open pair核公理，空carrier也有拓撲[[]]。", "en": "Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]." }, "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": "S004", "uses": [ "G001", "S003" ], "before": "ORIGINAL carrier order=[a20, b20, c20];source canonical open family=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []];actual open index=2,U=[a20, c20]", "after": "ALL [other_index,union,intersection] rows=[[0, [a20, b20, c20], [a20, c20]], [1, [a20, b20, c20], [a20]], [2, [a20, c20], [a20, c20]], [3, [a20, c20], [a20]], [4, [a20, b20, c20], [c20]], [5, [a20, b20, c20], []], [6, [a20, c20], [c20]], [7, [a20, c20], []]];empty/full present;family qualification passed with exact source English collation", "relation": "implies", "rule_id": "hausdorff_complete_labeled_topology_and_source_order", "conditions": [], "explanation": { "zh-Hant": "點標籤與原carrier次序保留；各open內依來源字串排序，family依固定English collation，不按大小或另一個常見慣例。逐全部open pair核公理，空carrier也有拓撲[[]]。", "en": "Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]." }, "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": "S005", "uses": [ "G001", "S004" ], "before": "ORIGINAL carrier order=[a20, b20, c20];source canonical open family=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []];actual open index=3,U=[a20]", "after": "ALL [other_index,union,intersection] rows=[[0, [a20, b20, c20], [a20]], [1, [a20, b20], [a20]], [2, [a20, c20], [a20]], [3, [a20], [a20]], [4, [a20, b20, c20], []], [5, [a20, b20], []], [6, [a20, c20], []], [7, [a20], []]];empty/full present;family qualification passed with exact source English collation", "relation": "implies", "rule_id": "hausdorff_complete_labeled_topology_and_source_order", "conditions": [], "explanation": { "zh-Hant": "點標籤與原carrier次序保留；各open內依來源字串排序，family依固定English collation，不按大小或另一個常見慣例。逐全部open pair核公理，空carrier也有拓撲[[]]。", "en": "Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]." }, "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": "S006", "uses": [ "G001", "S005" ], "before": "ORIGINAL carrier order=[a20, b20, c20];source canonical open family=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []];actual open index=4,U=[b20, c20]", "after": "ALL [other_index,union,intersection] rows=[[0, [a20, b20, c20], [b20, c20]], [1, [a20, b20, c20], [b20]], [2, [a20, b20, c20], [c20]], [3, [a20, b20, c20], []], [4, [b20, c20], [b20, c20]], [5, [b20, c20], [b20]], [6, [b20, c20], [c20]], [7, [b20, c20], []]];empty/full present;family qualification passed with exact source English collation", "relation": "implies", "rule_id": "hausdorff_complete_labeled_topology_and_source_order", "conditions": [], "explanation": { "zh-Hant": "點標籤與原carrier次序保留；各open內依來源字串排序，family依固定English collation，不按大小或另一個常見慣例。逐全部open pair核公理，空carrier也有拓撲[[]]。", "en": "Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]." }, "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": "S007", "uses": [ "G001", "S006" ], "before": "ORIGINAL carrier order=[a20, b20, c20];source canonical open family=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []];actual open index=5,U=[b20]", "after": "ALL [other_index,union,intersection] rows=[[0, [a20, b20, c20], [b20]], [1, [a20, b20], [b20]], [2, [a20, b20, c20], []], [3, [a20, b20], []], [4, [b20, c20], [b20]], [5, [b20], [b20]], [6, [b20, c20], []], [7, [b20], []]];empty/full present;family qualification passed with exact source English collation", "relation": "implies", "rule_id": "hausdorff_complete_labeled_topology_and_source_order", "conditions": [], "explanation": { "zh-Hant": "點標籤與原carrier次序保留；各open內依來源字串排序，family依固定English collation，不按大小或另一個常見慣例。逐全部open pair核公理，空carrier也有拓撲[[]]。", "en": "Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]." }, "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": "S008", "uses": [ "G001", "S007" ], "before": "ORIGINAL carrier order=[a20, b20, c20];source canonical open family=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []];actual open index=6,U=[c20]", "after": "ALL [other_index,union,intersection] rows=[[0, [a20, b20, c20], [c20]], [1, [a20, b20, c20], []], [2, [a20, c20], [c20]], [3, [a20, c20], []], [4, [b20, c20], [c20]], [5, [b20, c20], []], [6, [c20], [c20]], [7, [c20], []]];empty/full present;family qualification passed with exact source English collation", "relation": "implies", "rule_id": "hausdorff_complete_labeled_topology_and_source_order", "conditions": [], "explanation": { "zh-Hant": "點標籤與原carrier次序保留；各open內依來源字串排序，family依固定English collation，不按大小或另一個常見慣例。逐全部open pair核公理，空carrier也有拓撲[[]]。", "en": "Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "ORIGINAL carrier order=[a20, b20, c20];source canonical open family=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []];actual open index=7,U=[]", "after": "ALL [other_index,union,intersection] rows=[[0, [a20, b20, c20], []], [1, [a20, b20], []], [2, [a20, c20], []], [3, [a20], []], [4, [b20, c20], []], [5, [b20], []], [6, [c20], []], [7, [], []]];empty/full present;family qualification passed with exact source English collation", "relation": "implies", "rule_id": "hausdorff_complete_labeled_topology_and_source_order", "conditions": [], "explanation": { "zh-Hant": "點標籤與原carrier次序保留；各open內依來源字串排序，family依固定English collation，不按大小或另一個常見慣例。逐全部open pair核公理，空carrier也有拓撲[[]]。", "en": "Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]]." }, "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": "S010", "uses": [ "G001", "S009" ], "before": "Hausdorff ORIGINAL pair index0:ORIGINAL distinct carrier points x=a20,y=b20;source open order=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []]", "after": "ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a20, b20, c20], 否], [0, 1, [a20, b20], 否], [0, 4, [b20, c20], 否], [0, 5, [b20], 否], [1, 0, [a20, b20], 否], [1, 1, [a20, b20], 否], [1, 4, [b20], 否], [1, 5, [b20], 否], [2, 0, [a20, c20], 否], [2, 1, [a20], 否], [2, 4, [c20], 否], [2, 5, [], 是]];FIRST source-order witness=[[a20, c20], [b20]]", "relation": "implies", "rule_id": "hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure", "conditions": [], "explanation": { "zh-Hant": "只考慮U含x、V含y的真正open；按原U再V順序找第一個不交pair。null表示這個完整有限open-pair域沒有見證，不是抽樣沒看到。", "en": "Use actual opens U containing x and V containing y.Search original U then V order for the first disjoint pair.Null means no witness in this complete finite domain,not a failed sample." }, "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": "S011", "uses": [ "G001", "S010" ], "before": "Hausdorff ORIGINAL pair index1:ORIGINAL distinct carrier points x=a20,y=c20;source open order=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []]", "after": "ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a20, b20, c20], 否], [0, 2, [a20, c20], 否], [0, 4, [b20, c20], 否], [0, 6, [c20], 否], [1, 0, [a20, b20], 否], [1, 2, [a20], 否], [1, 4, [b20], 否], [1, 6, [], 是]];FIRST source-order witness=[[a20, b20], [c20]]", "relation": "implies", "rule_id": "hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure", "conditions": [], "explanation": { "zh-Hant": "只考慮U含x、V含y的真正open；按原U再V順序找第一個不交pair。null表示這個完整有限open-pair域沒有見證，不是抽樣沒看到。", "en": "Use actual opens U containing x and V containing y.Search original U then V order for the first disjoint pair.Null means no witness in this complete finite domain,not a failed sample." }, "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": "S012", "uses": [ "G001", "S011" ], "before": "Hausdorff ORIGINAL pair index2:ORIGINAL distinct carrier points x=b20,y=c20;source open order=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []]", "after": "ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a20, b20, c20], 否], [0, 2, [a20, c20], 否], [0, 4, [b20, c20], 否], [0, 6, [c20], 否], [1, 0, [a20, b20], 否], [1, 2, [a20], 否], [1, 4, [b20], 否], [1, 6, [], 是]];FIRST source-order witness=[[a20, b20], [c20]]", "relation": "implies", "rule_id": "hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure", "conditions": [], "explanation": { "zh-Hant": "只考慮U含x、V含y的真正open；按原U再V順序找第一個不交pair。null表示這個完整有限open-pair域沒有見證，不是抽樣沒看到。", "en": "Use actual opens U containing x and V containing y.Search original U then V order for the first disjoint pair.Null means no witness in this complete finite domain,not a failed sample." }, "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": "S013", "uses": [ "G001", "S012" ], "before": "ORIGINAL carrier order=[a20, b20, c20];ALL unordered distinct pairs with first index