第 22902 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

R0044-22902 · q-20261014-advance-20261111-r03-hausdorff-space-0027

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":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["b1","c1"],["b1"],["c1"],[]]}。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":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["b1","c1"],["b1"],["c1"],[]]}。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-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=[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

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

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

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

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

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

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

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=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

S011 · implies

操作前

Hausdorff ORIGINAL pair index1:ORIGINAL distinct carrier points x=a1,y=c1;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, 2, [a1, c1], 否], [0, 4, [b1, c1], 否], [0, 6, [c1], 否], [1, 0, [a1, b1], 否], [1, 2, [a1], 否], [1, 4, [b1], 否], [1, 6, [], 是]];FIRST source-order witness=[[a1, b1], [c1]]

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=b1,y=c1;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, 2, [a1, c1], 否], [0, 4, [b1, c1], 否], [0, 6, [c1], 否], [1, 0, [a1, b1], 否], [1, 2, [a1], 否], [1, 4, [b1], 否], [1, 6, [], 是]];FIRST source-order witness=[[a1, b1], [c1]]

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=[a1, b1, c1];ALL unordered distinct pairs with first index<second=[[a1, b1], [a1, c1], [b1, c1]]

操作後

ALL pair-witness rows=[[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]];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

([a1, b1, c1], [[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]], 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: 8eb5efbb83df3442b3d782dbc5ed33f24934a5fc08bfd40abba58f5845ab392f
candidate record SHA-256: 7a72801f3f6e26cdb2486d03e58bee407a7c99acda61438dde352be658fa6a4c
semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
Lean: NOT_RUN · human review: NOT_PERFORMED

同源完整 JSON