# 第 22959 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。

R0044-22959 · q-20261014-advance-20261111-r03-hausdorff-space-0009



## 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":["a8","b8","c8"],"topology":[["a8","b8","c8"],["a8","b8"],["a8","c8"],["a8"],["b8","c8"],["b8"],["c8"],[]],"x":"a8","y":"b8"}。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":["a8","b8","c8"],"topology":[["a8","b8","c8"],["a8","b8"],["a8","c8"],["a8"],["b8","c8"],["b8"],["c8"],[]],"x":"a8","y":"b8"}。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=[a8, b8, c8];source canonical open family=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []];actual open index=0,U=[a8, b8, c8]

操作後

ALL [other_index,union,intersection] rows=[[0, [a8, b8, c8], [a8, b8, c8]], [1, [a8, b8, c8], [a8, b8]], [2, [a8, b8, c8], [a8, c8]], [3, [a8, b8, c8], [a8]], [4, [a8, b8, c8], [b8, c8]], [5, [a8, b8, c8], [b8]], [6, [a8, b8, c8], [c8]], [7, [a8, b8, c8], []]];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=[a8, b8, c8];source canonical open family=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []];actual open index=1,U=[a8, b8]

操作後

ALL [other_index,union,intersection] rows=[[0, [a8, b8, c8], [a8, b8]], [1, [a8, b8], [a8, b8]], [2, [a8, b8, c8], [a8]], [3, [a8, b8], [a8]], [4, [a8, b8, c8], [b8]], [5, [a8, b8], [b8]], [6, [a8, b8, c8], []], [7, [a8, b8], []]];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=[a8, b8, c8];source canonical open family=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []];actual open index=2,U=[a8, c8]

操作後

ALL [other_index,union,intersection] rows=[[0, [a8, b8, c8], [a8, c8]], [1, [a8, b8, c8], [a8]], [2, [a8, c8], [a8, c8]], [3, [a8, c8], [a8]], [4, [a8, b8, c8], [c8]], [5, [a8, b8, c8], []], [6, [a8, c8], [c8]], [7, [a8, c8], []]];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=[a8, b8, c8];source canonical open family=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []];actual open index=3,U=[a8]

操作後

ALL [other_index,union,intersection] rows=[[0, [a8, b8, c8], [a8]], [1, [a8, b8], [a8]], [2, [a8, c8], [a8]], [3, [a8], [a8]], [4, [a8, b8, c8], []], [5, [a8, b8], []], [6, [a8, c8], []], [7, [a8], []]];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=[a8, b8, c8];source canonical open family=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []];actual open index=4,U=[b8, c8]

操作後

ALL [other_index,union,intersection] rows=[[0, [a8, b8, c8], [b8, c8]], [1, [a8, b8, c8], [b8]], [2, [a8, b8, c8], [c8]], [3, [a8, b8, c8], []], [4, [b8, c8], [b8, c8]], [5, [b8, c8], [b8]], [6, [b8, c8], [c8]], [7, [b8, c8], []]];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=[a8, b8, c8];source canonical open family=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []];actual open index=5,U=[b8]

操作後

ALL [other_index,union,intersection] rows=[[0, [a8, b8, c8], [b8]], [1, [a8, b8], [b8]], [2, [a8, b8, c8], []], [3, [a8, b8], []], [4, [b8, c8], [b8]], [5, [b8], [b8]], [6, [b8, c8], []], [7, [b8], []]];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=[a8, b8, c8];source canonical open family=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []];actual open index=6,U=[c8]

操作後

ALL [other_index,union,intersection] rows=[[0, [a8, b8, c8], [c8]], [1, [a8, b8, c8], []], [2, [a8, c8], [c8]], [3, [a8, c8], []], [4, [b8, c8], [c8]], [5, [b8, c8], []], [6, [c8], [c8]], [7, [c8], []]];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=[a8, b8, c8];source canonical open family=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []];actual open index=7,U=[]

操作後

ALL [other_index,union,intersection] rows=[[0, [a8, b8, c8], []], [1, [a8, b8], []], [2, [a8, c8], []], [3, [a8], []], [4, [b8, c8], []], [5, [b8], []], [6, [c8], []], [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=a8,y=b8;source open order=[[a8, b8, c8], [a8, b8], [a8, c8], [a8], [b8, c8], [b8], [c8], []]

操作後

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a8, b8, c8], 否], [0, 1, [a8, b8], 否], [0, 4, [b8, c8], 否], [0, 5, [b8], 否], [1, 0, [a8, b8], 否], [1, 1, [a8, b8], 否], [1, 4, [b8], 否], [1, 5, [b8], 否], [2, 0, [a8, c8], 否], [2, 1, [a8], 否], [2, 4, [c8], 否], [2, 5, [], 是]];FIRST source-order witness=[[a8, c8], [b8]]

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

(a8, b8, [[a8, c8], [b8]], 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: 897c09a005df3165d1f0980fefc6394d24427dc52f06ef908490a56aa26d333c
- candidate record SHA-256: cc314a50cdf82368c1f2c984887ac94c960a93141ab6ee8e85e5a6e6ec9aea41
- 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-0009.json)