第 22941 題：Hausdorff空間

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原題與已知條件
定義與適用範圍
逐步推導
逐步反向核對
結論
本題覆核與來源綁定

R0044-22941 · q-20261014-advance-20261111-r03-hausdorff-space-0066

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-inseparable-pair；資料={"points":["a15","b15","c15"],"topology":[["a15","b15","c15"],["a15","b15"],["a15"],[]],"x":"a15","y":"b15"}。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-inseparable-pair；資料={"points":["a15","b15","c15"],"topology":[["a15","b15","c15"],["a15","b15"],["a15"],[]],"x":"a15","y":"b15"}。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

逐步推導

S001 · implies

操作前

source operation=hausdorff-inseparable-pair;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[x, y, witness_or_null, separated, boundary]

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=[a15, b15, c15];source canonical open family=[[a15, b15, c15], [a15, b15], [a15], []];actual open index=0,U=[a15, b15, c15]

操作後

ALL [other_index,union,intersection] rows=[[0, [a15, b15, c15], [a15, b15, c15]], [1, [a15, b15, c15], [a15, b15]], [2, [a15, b15, c15], [a15]], [3, [a15, b15, c15], []]];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=[a15, b15, c15];source canonical open family=[[a15, b15, c15], [a15, b15], [a15], []];actual open index=1,U=[a15, b15]

操作後

ALL [other_index,union,intersection] rows=[[0, [a15, b15, c15], [a15, b15]], [1, [a15, b15], [a15, b15]], [2, [a15, b15], [a15]], [3, [a15, b15], []]];empty/full present;family qualification passed with exact source English collation

條件：無

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=[a15, b15, c15];source canonical open family=[[a15, b15, c15], [a15, b15], [a15], []];actual open index=2,U=[a15]

操作後

ALL [other_index,union,intersection] rows=[[0, [a15, b15, c15], [a15]], [1, [a15, b15], [a15]], [2, [a15], [a15]], [3, [a15], []]];empty/full present;family qualification passed with exact source English collation

條件：無

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=[a15, b15, c15];source canonical open family=[[a15, b15, c15], [a15, b15], [a15], []];actual open index=3,U=[]

操作後

ALL [other_index,union,intersection] rows=[[0, [a15, b15, c15], []], [1, [a15, b15], []], [2, [a15], []], [3, [], []]];empty/full present;family qualification passed with exact source English collation

條件：無

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

Hausdorff ORIGINAL designated pair:ORIGINAL distinct carrier points x=a15,y=b15;source open order=[[a15, b15, c15], [a15, b15], [a15], []]

操作後

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a15, b15, c15], 否], [0, 1, [a15, b15], 否], [1, 0, [a15, b15], 否], [1, 1, [a15, b15], 否], [2, 0, [a15], 否], [2, 1, [a15], 否]];FIRST source-order witness=null

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, S005

逐步反向核對

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

(a15, b15, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)

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: 3ccd33c15029dd68aaeece2b92f8e9d6d80ba5f292bf6923872bebce5a8221f6
candidate record SHA-256: 16c2fcaa498c2a72534358afc5deaec39f70d0641b26a24b521a9f9925f53d13
semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
Lean: NOT_RUN · human review: NOT_PERFORMED

同源完整 JSON