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第 22975 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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R0044-22975 · q-20261014-advance-20261111-r03-hausdorff-space-0025

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-pair-separator；資料={"points":["a24","b24","c24"],"topology":[["a24","b24","c24"],["a24","b24"],["a24","c24"],["a24"],["b24","c24"],["b24"],["c24"],[]],"x":"a24","y":"b24"}。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-pair-separator；資料={"points":["a24","b24","c24"],"topology":[["a24","b24","c24"],["a24","b24"],["a24","c24"],["a24"],["b24","c24"],["b24"],["c24"],[]],"x":"a24","y":"b24"}。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-pair-separator;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[x, y, open_neighborhood_witness, separated, witness_disjoint]

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=[a24, b24, c24];source canonical open family=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []];actual open index=0,U=[a24, b24, c24]

操作後

ALL [other_index,union,intersection] rows=[[0, [a24, b24, c24], [a24, b24, c24]], [1, [a24, b24, c24], [a24, b24]], [2, [a24, b24, c24], [a24, c24]], [3, [a24, b24, c24], [a24]], [4, [a24, b24, c24], [b24, c24]], [5, [a24, b24, c24], [b24]], [6, [a24, b24, c24], [c24]], [7, [a24, b24, c24], []]];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=[a24, b24, c24];source canonical open family=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []];actual open index=1,U=[a24, b24]

操作後

ALL [other_index,union,intersection] rows=[[0, [a24, b24, c24], [a24, b24]], [1, [a24, b24], [a24, b24]], [2, [a24, b24, c24], [a24]], [3, [a24, b24], [a24]], [4, [a24, b24, c24], [b24]], [5, [a24, b24], [b24]], [6, [a24, b24, c24], []], [7, [a24, b24], []]];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=[a24, b24, c24];source canonical open family=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []];actual open index=2,U=[a24, c24]

操作後

ALL [other_index,union,intersection] rows=[[0, [a24, b24, c24], [a24, c24]], [1, [a24, b24, c24], [a24]], [2, [a24, c24], [a24, c24]], [3, [a24, c24], [a24]], [4, [a24, b24, c24], [c24]], [5, [a24, b24, c24], []], [6, [a24, c24], [c24]], [7, [a24, c24], []]];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=[a24, b24, c24];source canonical open family=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []];actual open index=3,U=[a24]

操作後

ALL [other_index,union,intersection] rows=[[0, [a24, b24, c24], [a24]], [1, [a24, b24], [a24]], [2, [a24, c24], [a24]], [3, [a24], [a24]], [4, [a24, b24, c24], []], [5, [a24, b24], []], [6, [a24, c24], []], [7, [a24], []]];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

ORIGINAL carrier order=[a24, b24, c24];source canonical open family=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []];actual open index=4,U=[b24, c24]

操作後

ALL [other_index,union,intersection] rows=[[0, [a24, b24, c24], [b24, c24]], [1, [a24, b24, c24], [b24]], [2, [a24, b24, c24], [c24]], [3, [a24, b24, c24], []], [4, [b24, c24], [b24, c24]], [5, [b24, c24], [b24]], [6, [b24, c24], [c24]], [7, [b24, c24], []]];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, S005

S007 · implies

操作前

ORIGINAL carrier order=[a24, b24, c24];source canonical open family=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []];actual open index=5,U=[b24]

操作後

ALL [other_index,union,intersection] rows=[[0, [a24, b24, c24], [b24]], [1, [a24, b24], [b24]], [2, [a24, b24, c24], []], [3, [a24, b24], []], [4, [b24, c24], [b24]], [5, [b24], [b24]], [6, [b24, c24], []], [7, [b24], []]];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, S006

S008 · implies

操作前

ORIGINAL carrier order=[a24, b24, c24];source canonical open family=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []];actual open index=6,U=[c24]

操作後

ALL [other_index,union,intersection] rows=[[0, [a24, b24, c24], [c24]], [1, [a24, b24, c24], []], [2, [a24, c24], [c24]], [3, [a24, c24], []], [4, [b24, c24], [c24]], [5, [b24, c24], []], [6, [c24], [c24]], [7, [c24], []]];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, S007

S009 · implies

操作前

ORIGINAL carrier order=[a24, b24, c24];source canonical open family=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []];actual open index=7,U=[]

操作後

ALL [other_index,union,intersection] rows=[[0, [a24, b24, c24], []], [1, [a24, b24], []], [2, [a24, c24], []], [3, [a24], []], [4, [b24, c24], []], [5, [b24], []], [6, [c24], []], [7, [], []]];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, S008

S010 · implies

操作前

Hausdorff ORIGINAL designated pair:ORIGINAL distinct carrier points x=a24,y=b24;source open order=[[a24, b24, c24], [a24, b24], [a24, c24], [a24], [b24, c24], [b24], [c24], []]

操作後

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a24, b24, c24], 否], [0, 1, [a24, b24], 否], [0, 4, [b24, c24], 否], [0, 5, [b24], 否], [1, 0, [a24, b24], 否], [1, 1, [a24, b24], 否], [1, 4, [b24], 否], [1, 5, [b24], 否], [2, 0, [a24, c24], 否], [2, 1, [a24], 否], [2, 4, [c24], 否], [2, 5, [], 是]];FIRST source-order witness=[[a24, c24], [b24]]

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, S009

逐步反向核對

  • S010 · 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。
  • S009 · 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。
  • S008 · 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。
  • S007 · 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。
  • 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

(a24, b24, [[a24, c24], [b24]], 是, 是)

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: 401468148180919f96a573614d913994a7ccb1d14294acd6c26dba0023477a71
  • candidate record SHA-256: ec5c3abda8ba55008358b484db81787318396461215884e36ca1c7b94e3b2e9f
  • semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
  • question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON