# 第 22952 題：Hausdorff空間

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本頁是已完成第二對話 AI 數理與中英語義覆核的教學詳解，只支持本題明示的模型與前提。沒有人工審查或新 Lean 證明，也不提升原題或 MKO 的 Canonical 真值狀態。

R0044-22952 · q-20261014-advance-20261111-r03-hausdorff-space-0002



## 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":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["b1","c1"],["b1"],["c1"],[]],"x":"a1","y":"b1"}。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":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["b1","c1"],["b1"],["c1"],[]],"x":"a1","y":"b1"}。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。


- [mko-hausdorff-space](https://ocme.evemisslab.com/zh-Hant/mko/mko-hausdorff-space/)





## 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=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=0,U=[a1, b1, c1]

操作後

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [a1, b1, c1]], [1, [a1, b1, c1], [a1, b1]], [2, [a1, b1, c1], [a1, c1]], [3, [a1, b1, c1], [a1]], [4, [a1, b1, c1], [b1, c1]], [5, [a1, b1, c1], [b1]], [6, [a1, b1, c1], [c1]], [7, [a1, b1, c1], []]];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=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=1,U=[a1, b1]

操作後

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [a1, b1]], [1, [a1, b1], [a1, b1]], [2, [a1, b1, c1], [a1]], [3, [a1, b1], [a1]], [4, [a1, b1, c1], [b1]], [5, [a1, b1], [b1]], [6, [a1, b1, c1], []], [7, [a1, b1], []]];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=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=2,U=[a1, c1]

操作後

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [a1, c1]], [1, [a1, b1, c1], [a1]], [2, [a1, c1], [a1, c1]], [3, [a1, c1], [a1]], [4, [a1, b1, c1], [c1]], [5, [a1, b1, c1], []], [6, [a1, c1], [c1]], [7, [a1, c1], []]];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=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=3,U=[a1]

操作後

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [a1]], [1, [a1, b1], [a1]], [2, [a1, c1], [a1]], [3, [a1], [a1]], [4, [a1, b1, c1], []], [5, [a1, b1], []], [6, [a1, c1], []], [7, [a1], []]];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=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=4,U=[b1, c1]

操作後

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [b1, c1]], [1, [a1, b1, c1], [b1]], [2, [a1, b1, c1], [c1]], [3, [a1, b1, c1], []], [4, [b1, c1], [b1, c1]], [5, [b1, c1], [b1]], [6, [b1, c1], [c1]], [7, [b1, c1], []]];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=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=5,U=[b1]

操作後

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [b1]], [1, [a1, b1], [b1]], [2, [a1, b1, c1], []], [3, [a1, b1], []], [4, [b1, c1], [b1]], [5, [b1], [b1]], [6, [b1, c1], []], [7, [b1], []]];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=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=6,U=[c1]

操作後

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [c1]], [1, [a1, b1, c1], []], [2, [a1, c1], [c1]], [3, [a1, c1], []], [4, [b1, c1], [c1]], [5, [b1, c1], []], [6, [c1], [c1]], [7, [c1], []]];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=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=7,U=[]

操作後

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], []], [1, [a1, b1], []], [2, [a1, c1], []], [3, [a1], []], [4, [b1, c1], []], [5, [b1], []], [6, [c1], []], [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=a1,y=b1;source open order=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []]

操作後

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a1, b1, c1], 否], [0, 1, [a1, b1], 否], [0, 4, [b1, c1], 否], [0, 5, [b1], 否], [1, 0, [a1, b1], 否], [1, 1, [a1, b1], 否], [1, 4, [b1], 否], [1, 5, [b1], 否], [2, 0, [a1, c1], 否], [2, 1, [a1], 否], [2, 4, [c1], 否], [2, 5, [], 是]];FIRST source-order witness=[[a1, c1], [b1]]

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

(a1, b1, [[a1, c1], [b1]], 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: 3e13ff0573a3aadc57324cdc8601680ac2a7bcac9dea80ac9519a22209296745
- candidate record SHA-256: eb6e23ad894ea20e265b0f2c792f77a455ffff9f997bae2b7c17e7a01e95ce6f
- semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
- question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
- Lean: NOT_RUN · human review: NOT_PERFORMED

[同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r03-hausdorff-space-0002.json)