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第 22947 題：Hausdorff空間

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  1. 原題與已知條件
  2. 定義與適用範圍
  3. 逐步推導
  4. 逐步反向核對
  5. 結論
  6. 本題覆核與來源綁定

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R0044-22947 · q-20261014-advance-20261111-r03-hausdorff-space-0072

原題與已知條件

完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={"points":["a21","b21","c21"],"topology":[["a21","b21","c21"],["a21","b21"],["a21"],[]],"x":"a21","y":"b21"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

  • G001 — 完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={"points":["a21","b21","c21"],"topology":[["a21","b21","c21"],["a21","b21"],["a21"],[]],"x":"a21","y":"b21"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

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。

逐步推導

S001 · implies

操作前

source operation=hausdorff-inseparable-pair;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[x, y, witness_or_null, separated, boundary]

條件：無

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。

步驟識別與依賴

規則：source_ordered_interface_explicit_without_javascript · 依賴：G001

S002 · implies

操作前

ORIGINAL carrier order=[a21, b21, c21];source canonical open family=[[a21, b21, c21], [a21, b21], [a21], []];actual open index=0,U=[a21, b21, c21]

操作後

ALL [other_index,union,intersection] rows=[[0, [a21, b21, c21], [a21, b21, c21]], [1, [a21, b21, c21], [a21, b21]], [2, [a21, b21, c21], [a21]], [3, [a21, b21, c21], []]];empty/full present;family qualification passed with exact source English collation

條件：無

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[[]]。

步驟識別與依賴

規則：hausdorff_complete_labeled_topology_and_source_order · 依賴：G001, S001

S003 · implies

操作前

ORIGINAL carrier order=[a21, b21, c21];source canonical open family=[[a21, b21, c21], [a21, b21], [a21], []];actual open index=1,U=[a21, b21]

操作後

ALL [other_index,union,intersection] rows=[[0, [a21, b21, c21], [a21, b21]], [1, [a21, b21], [a21, b21]], [2, [a21, b21], [a21]], [3, [a21, b21], []]];empty/full present;family qualification passed with exact source English collation

條件：無

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[[]]。

步驟識別與依賴

規則：hausdorff_complete_labeled_topology_and_source_order · 依賴：G001, S002

S004 · implies

操作前

ORIGINAL carrier order=[a21, b21, c21];source canonical open family=[[a21, b21, c21], [a21, b21], [a21], []];actual open index=2,U=[a21]

操作後

ALL [other_index,union,intersection] rows=[[0, [a21, b21, c21], [a21]], [1, [a21, b21], [a21]], [2, [a21], [a21]], [3, [a21], []]];empty/full present;family qualification passed with exact source English collation

條件：無

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[[]]。

步驟識別與依賴

規則：hausdorff_complete_labeled_topology_and_source_order · 依賴：G001, S003

S005 · implies

操作前

ORIGINAL carrier order=[a21, b21, c21];source canonical open family=[[a21, b21, c21], [a21, b21], [a21], []];actual open index=3,U=[]

操作後

ALL [other_index,union,intersection] rows=[[0, [a21, b21, c21], []], [1, [a21, b21], []], [2, [a21], []], [3, [], []]];empty/full present;family qualification passed with exact source English collation

條件：無

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[[]]。

步驟識別與依賴

規則：hausdorff_complete_labeled_topology_and_source_order · 依賴：G001, S004

S006 · implies

操作前

Hausdorff ORIGINAL designated pair:ORIGINAL distinct carrier points x=a21,y=b21;source open order=[[a21, b21, c21], [a21, b21], [a21], []]

操作後

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a21, b21, c21], 否], [0, 1, [a21, b21], 否], [1, 0, [a21, b21], 否], [1, 1, [a21, b21], 否], [2, 0, [a21], 否], [2, 1, [a21], 否]];FIRST source-order witness=null

條件：無

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。

步驟識別與依賴

規則：hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure · 依賴：G001, S005

逐步反向核對

  • S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
  • S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
  • S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
  • S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
  • S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
  • S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(a21, b21, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)

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。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。

來源雜湊與覆核憑證
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 6596c86a3fb836394aa785edf5611246eb71e1987d48aa5f5b3b715cdd6f387f
  • candidate record SHA-256: 7a854b72e18e3ebd055bf8c819d05f83d430735f1809a459f0b1f5cfc7abf924
  • semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
  • question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON