# 第 22988 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。

R0044-22988 · q-20261014-advance-20261111-r03-hausdorff-space-0088



## 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-sampled-boundary；資料={"points":["a12","b12","c12"],"topology":[["a12","b12","c12"],["a12","b12"],["a12","c12"],["a12"],["c12"],[]],"sampled_pairs":[["c12","a12"]]}。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-sampled-boundary；資料={"points":["a12","b12","c12"],"topology":[["a12","b12","c12"],["a12","b12"],["a12","c12"],["a12"],["c12"],[]],"sampled_pairs":[["c12","a12"]]}。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-sampled-boundary;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, 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=[a12, b12, c12];source canonical open family=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [c12], []];actual open index=0,U=[a12, b12, c12]

操作後

ALL [other_index,union,intersection] rows=[[0, [a12, b12, c12], [a12, b12, c12]], [1, [a12, b12, c12], [a12, b12]], [2, [a12, b12, c12], [a12, c12]], [3, [a12, b12, c12], [a12]], [4, [a12, b12, c12], [c12]], [5, [a12, b12, c12], []]];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=[a12, b12, c12];source canonical open family=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [c12], []];actual open index=1,U=[a12, b12]

操作後

ALL [other_index,union,intersection] rows=[[0, [a12, b12, c12], [a12, b12]], [1, [a12, b12], [a12, b12]], [2, [a12, b12, c12], [a12]], [3, [a12, b12], [a12]], [4, [a12, b12, c12], []], [5, [a12, b12], []]];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=[a12, b12, c12];source canonical open family=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [c12], []];actual open index=2,U=[a12, c12]

操作後

ALL [other_index,union,intersection] rows=[[0, [a12, b12, c12], [a12, c12]], [1, [a12, b12, c12], [a12]], [2, [a12, c12], [a12, c12]], [3, [a12, c12], [a12]], [4, [a12, c12], [c12]], [5, [a12, c12], []]];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=[a12, b12, c12];source canonical open family=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [c12], []];actual open index=3,U=[a12]

操作後

ALL [other_index,union,intersection] rows=[[0, [a12, b12, c12], [a12]], [1, [a12, b12], [a12]], [2, [a12, c12], [a12]], [3, [a12], [a12]], [4, [a12, c12], []], [5, [a12], []]];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=[a12, b12, c12];source canonical open family=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [c12], []];actual open index=4,U=[c12]

操作後

ALL [other_index,union,intersection] rows=[[0, [a12, b12, c12], [c12]], [1, [a12, b12, c12], []], [2, [a12, c12], [c12]], [3, [a12, c12], []], [4, [c12], [c12]], [5, [c12], []]];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=[a12, b12, c12];source canonical open family=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [c12], []];actual open index=5,U=[]

操作後

ALL [other_index,union,intersection] rows=[[0, [a12, b12, c12], []], [1, [a12, b12], []], [2, [a12, c12], []], [3, [a12], []], [4, [c12], []], [5, [], []]];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

Hausdorff ORIGINAL sampled pair index0:ORIGINAL distinct carrier points x=c12,y=a12;source open order=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [c12], []]

操作後

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a12, b12, c12], 否], [0, 1, [a12, b12], 否], [0, 2, [a12, c12], 否], [0, 3, [a12], 否], [2, 0, [a12, c12], 否], [2, 1, [a12], 否], [2, 2, [a12, c12], 否], [2, 3, [a12], 否], [4, 0, [c12], 否], [4, 1, [], 是]];FIRST source-order witness=[[c12], [a12, b12]]

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, S007





S009 · calculation操作前

Hausdorff ORIGINAL sampled positions count := sum(1)

操作後

Hausdorff ORIGINAL sampled positions count = 1

條件：無

依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。

步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S008





S010 · calculation操作前

Hausdorff ALL unordered carrier pairs count := sum(1, 1, 1)

操作後

Hausdorff ALL unordered carrier pairs count = 3

條件：無

依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。

步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S009





S011 · implies操作前

ORIGINAL sampled pair rows=[[c12, a12, [[c12], [a12, b12]]]];sample positions count=1;ALL finite pairs=[[a12, b12], [a12, c12], [b12, c12]];total pair count=3

操作後

all_sampled_separated=是;general_sample_certificate fixed No;duplicates/full finite sample allowed;any actual completeness must be established separately

條件：無

sample可空、重複或真的覆蓋全部有限pair；位置數不等於不同pair數。固定No不否定各pair的真分離，也不否認另核完整性後的有限結論，更不從有限抽樣推無窮空間。

步驟識別與依賴規則：hausdorff_sample_No_not_separation_flag_or_actual_completeness_negation · 依賴：G001, S010









## 逐步反向核對


- 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 — 以精確有理數重算 Hausdorff ALL unordered carrier pairs count，不以小數近似替代。
- S009 · premise_and_result_audit — 以精確有理數重算 Hausdorff ORIGINAL sampled positions count，不以小數近似替代。
- 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

([[c12, a12, [[c12], [a12, b12]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

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: 4da7bea9a8d9cd73bbb80b9aa8844c72f0843669ecd4ccbe8bf62a32a42171a4
- candidate record SHA-256: d8ee5893c1a3e9371f6a3411c0e35ffc937dedaa74bac4a610d57f654a674887
- 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-0088.json)