# 第 22919 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。

R0044-22919 · q-20261014-advance-20261111-r03-hausdorff-space-0044



## 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":["a18","b18","c18"],"topology":[["a18","b18","c18"],["a18","b18"],["a18","c18"],["a18"],["b18","c18"],["b18"],["c18"],[]]}。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":["a18","b18","c18"],"topology":[["a18","b18","c18"],["a18","b18"],["a18","c18"],["a18"],["b18","c18"],["b18"],["c18"],[]]}。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=[a18, b18, c18];source canonical open family=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []];actual open index=0,U=[a18, b18, c18]

操作後

ALL [other_index,union,intersection] rows=[[0, [a18, b18, c18], [a18, b18, c18]], [1, [a18, b18, c18], [a18, b18]], [2, [a18, b18, c18], [a18, c18]], [3, [a18, b18, c18], [a18]], [4, [a18, b18, c18], [b18, c18]], [5, [a18, b18, c18], [b18]], [6, [a18, b18, c18], [c18]], [7, [a18, b18, c18], []]];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=[a18, b18, c18];source canonical open family=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []];actual open index=1,U=[a18, b18]

操作後

ALL [other_index,union,intersection] rows=[[0, [a18, b18, c18], [a18, b18]], [1, [a18, b18], [a18, b18]], [2, [a18, b18, c18], [a18]], [3, [a18, b18], [a18]], [4, [a18, b18, c18], [b18]], [5, [a18, b18], [b18]], [6, [a18, b18, c18], []], [7, [a18, b18], []]];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=[a18, b18, c18];source canonical open family=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []];actual open index=2,U=[a18, c18]

操作後

ALL [other_index,union,intersection] rows=[[0, [a18, b18, c18], [a18, c18]], [1, [a18, b18, c18], [a18]], [2, [a18, c18], [a18, c18]], [3, [a18, c18], [a18]], [4, [a18, b18, c18], [c18]], [5, [a18, b18, c18], []], [6, [a18, c18], [c18]], [7, [a18, c18], []]];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=[a18, b18, c18];source canonical open family=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []];actual open index=3,U=[a18]

操作後

ALL [other_index,union,intersection] rows=[[0, [a18, b18, c18], [a18]], [1, [a18, b18], [a18]], [2, [a18, c18], [a18]], [3, [a18], [a18]], [4, [a18, b18, c18], []], [5, [a18, b18], []], [6, [a18, c18], []], [7, [a18], []]];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=[a18, b18, c18];source canonical open family=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []];actual open index=4,U=[b18, c18]

操作後

ALL [other_index,union,intersection] rows=[[0, [a18, b18, c18], [b18, c18]], [1, [a18, b18, c18], [b18]], [2, [a18, b18, c18], [c18]], [3, [a18, b18, c18], []], [4, [b18, c18], [b18, c18]], [5, [b18, c18], [b18]], [6, [b18, c18], [c18]], [7, [b18, c18], []]];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=[a18, b18, c18];source canonical open family=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []];actual open index=5,U=[b18]

操作後

ALL [other_index,union,intersection] rows=[[0, [a18, b18, c18], [b18]], [1, [a18, b18], [b18]], [2, [a18, b18, c18], []], [3, [a18, b18], []], [4, [b18, c18], [b18]], [5, [b18], [b18]], [6, [b18, c18], []], [7, [b18], []]];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=[a18, b18, c18];source canonical open family=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []];actual open index=6,U=[c18]

操作後

ALL [other_index,union,intersection] rows=[[0, [a18, b18, c18], [c18]], [1, [a18, b18, c18], []], [2, [a18, c18], [c18]], [3, [a18, c18], []], [4, [b18, c18], [c18]], [5, [b18, c18], []], [6, [c18], [c18]], [7, [c18], []]];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=[a18, b18, c18];source canonical open family=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []];actual open index=7,U=[]

操作後

ALL [other_index,union,intersection] rows=[[0, [a18, b18, c18], []], [1, [a18, b18], []], [2, [a18, c18], []], [3, [a18], []], [4, [b18, c18], []], [5, [b18], []], [6, [c18], []], [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=a18,y=b18;source open order=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []]

操作後

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a18, b18, c18], 否], [0, 1, [a18, b18], 否], [0, 4, [b18, c18], 否], [0, 5, [b18], 否], [1, 0, [a18, b18], 否], [1, 1, [a18, b18], 否], [1, 4, [b18], 否], [1, 5, [b18], 否], [2, 0, [a18, c18], 否], [2, 1, [a18], 否], [2, 4, [c18], 否], [2, 5, [], 是]];FIRST source-order witness=[[a18, c18], [b18]]

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=a18,y=c18;source open order=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []]

操作後

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a18, b18, c18], 否], [0, 2, [a18, c18], 否], [0, 4, [b18, c18], 否], [0, 6, [c18], 否], [1, 0, [a18, b18], 否], [1, 2, [a18], 否], [1, 4, [b18], 否], [1, 6, [], 是]];FIRST source-order witness=[[a18, b18], [c18]]

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=b18,y=c18;source open order=[[a18, b18, c18], [a18, b18], [a18, c18], [a18], [b18, c18], [b18], [c18], []]

操作後

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a18, b18, c18], 否], [0, 2, [a18, c18], 否], [0, 4, [b18, c18], 否], [0, 6, [c18], 否], [1, 0, [a18, b18], 否], [1, 2, [a18], 否], [1, 4, [b18], 否], [1, 6, [], 是]];FIRST source-order witness=[[a18, b18], [c18]]

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=[a18, b18, c18];ALL unordered distinct pairs with first index<second=[[a18, b18], [a18, c18], [b18, c18]]

操作後

ALL pair-witness rows=[[a18, b18, [[a18, c18], [b18]]], [a18, c18, [[a18, b18], [c18]]], [b18, c18, [[a18, b18], [c18]]]];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

([a18, b18, c18], [[a18, b18, [[a18, c18], [b18]]], [a18, c18, [[a18, b18], [c18]]], [b18, c18, [[a18, b18], [c18]]]], 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: 90098ced916be2dd6c53367a1b5601a66b3765fda2acda8a431261c0e27cb9fc
- candidate record SHA-256: 3ce93ce846c99ad7b60517df6e35be3c9abc492b03cae70f9a91a01a15b6c29e
- 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-0044.json)