# 第 22921 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。

R0044-22921 · q-20261014-advance-20261111-r03-hausdorff-space-0046



## 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-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。


- G001 — 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。





## 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-full-audit;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[points, pair_witness_rows, Hausdorff, certificate]

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=[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]

操作後

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

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=[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]

操作後

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

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=[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]

操作後

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

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=[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]

操作後

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

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=[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]

操作後

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

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=[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]

操作後

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

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=[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]

操作後

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

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=[a20, b20, c20];source canonical open family=[[a20, b20, c20], [a20, b20], [a20, c20], [a20], [b20, c20], [b20], [c20], []];actual open index=7,U=[]

操作後

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

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 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], []]

操作後

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]]

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





S011 · implies操作前

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], []]

操作後

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]]

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, S010





S012 · implies操作前

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], []]

操作後

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]]

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, S011





S013 · implies操作前

ORIGINAL carrier order=[a20, b20, c20];ALL unordered distinct pairs with first index<second=[[a20, b20], [a20, c20], [b20, c20]]

操作後

ALL pair-witness rows=[[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]];Hausdorff=是;no pairs means vacuous True,not failure

條件：無

一個完整失敗pair即可反駁Hausdorff；所有pair成功才給此有限空間證書。有限Hausdorff時，固定x對每個其他y取排除y的x鄰域，有限交便是{x} open，所以discrete；不能外推無窮空間。

步驟識別與依賴規則：hausdorff_ALL_original_point_pairs_empty_singleton_vacuity · 依賴：G001, S012









## 逐步反向核對


- S013 · 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。
- S012 · 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。
- S011 · 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。
- 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

([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: 3c628409a9a487bd476cea557a3bcf9a2d4dcfddc49badf15dd7bcc102b95abb
- candidate record SHA-256: f5ca09264e723706ad7820045411a4ed6c9c1b2f59afce98cd65a1534f14d85f
- 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-0046.json)