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第 22977 題：Hausdorff空間

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

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← 逐題詳解目錄 · 原題批次

R0044-22977 · q-20261014-advance-20261111-r03-hausdorff-space-0077

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":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["c1"],[]],"sampled_pairs":[["c1","a1"]]}。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":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["c1"],[]],"sampled_pairs":[["c1","a1"]]}。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。

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=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [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], [c1]], [5, [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], [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], []], [5, [a1, b1], []]];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=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [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, c1], [c1]], [5, [a1, 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, S003

S005 · implies

操作前

ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [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, c1], []], [5, [a1], []]];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=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [c1], []];actual open index=4,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, [c1], [c1]], [5, [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, S005

S007 · implies

操作前

ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [c1], []];actual open index=5,U=[]

操作後

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], []], [1, [a1, b1], []], [2, [a1, c1], []], [3, [a1], []], [4, [c1], []], [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=c1,y=a1;source open order=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [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, 2, [a1, c1], 否], [0, 3, [a1], 否], [2, 0, [a1, c1], 否], [2, 1, [a1], 否], [2, 2, [a1, c1], 否], [2, 3, [a1], 否], [4, 0, [c1], 否], [4, 1, [], 是]];FIRST source-order witness=[[c1], [a1, 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, 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=[[c1, a1, [[c1], [a1, b1]]]];sample positions count=1;ALL finite pairs=[[a1, b1], [a1, c1], [b1, c1]];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

([[c1, a1, [[c1], [a1, b1]]]], 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: b5938fea990f120de663a7da39c504bc590735ac0d7b162dc7982548b1304df9
  • candidate record SHA-256: 94ce48d0a6073c46943570d8417700a899d0a98e48fc2f76726c68a8ee69c35c
  • semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
  • question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON