第 22974 題：Hausdorff空間

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":["a23","b23","c23"],"topology":[["a23","b23","c23"],["a23","b23"],["a23","c23"],["a23"],["b23","c23"],["b23"],["c23"],[]],"x":"a23","y":"b23"}。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":["a23","b23","c23"],"topology":[["a23","b23","c23"],["a23","b23"],["a23","c23"],["a23"],["b23","c23"],["b23"],["c23"],[]],"x":"a23","y":"b23"}。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

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

explicit ordered source output fields=[x, y, open_neighborhood_witness, separated, witness_disjoint]

ORIGINAL carrier order=[a23, b23, c23];source canonical open family=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []];actual open index=0,U=[a23, b23, c23]

ALL [other_index,union,intersection] rows=[[0, [a23, b23, c23], [a23, b23, c23]], [1, [a23, b23, c23], [a23, b23]], [2, [a23, b23, c23], [a23, c23]], [3, [a23, b23, c23], [a23]], [4, [a23, b23, c23], [b23, c23]], [5, [a23, b23, c23], [b23]], [6, [a23, b23, c23], [c23]], [7, [a23, b23, c23], []]];empty/full present;family qualification passed with exact source English collation

ORIGINAL carrier order=[a23, b23, c23];source canonical open family=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []];actual open index=1,U=[a23, b23]

ALL [other_index,union,intersection] rows=[[0, [a23, b23, c23], [a23, b23]], [1, [a23, b23], [a23, b23]], [2, [a23, b23, c23], [a23]], [3, [a23, b23], [a23]], [4, [a23, b23, c23], [b23]], [5, [a23, b23], [b23]], [6, [a23, b23, c23], []], [7, [a23, b23], []]];empty/full present;family qualification passed with exact source English collation

ORIGINAL carrier order=[a23, b23, c23];source canonical open family=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []];actual open index=2,U=[a23, c23]

ALL [other_index,union,intersection] rows=[[0, [a23, b23, c23], [a23, c23]], [1, [a23, b23, c23], [a23]], [2, [a23, c23], [a23, c23]], [3, [a23, c23], [a23]], [4, [a23, b23, c23], [c23]], [5, [a23, b23, c23], []], [6, [a23, c23], [c23]], [7, [a23, c23], []]];empty/full present;family qualification passed with exact source English collation

ORIGINAL carrier order=[a23, b23, c23];source canonical open family=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []];actual open index=3,U=[a23]

ALL [other_index,union,intersection] rows=[[0, [a23, b23, c23], [a23]], [1, [a23, b23], [a23]], [2, [a23, c23], [a23]], [3, [a23], [a23]], [4, [a23, b23, c23], []], [5, [a23, b23], []], [6, [a23, c23], []], [7, [a23], []]];empty/full present;family qualification passed with exact source English collation

ORIGINAL carrier order=[a23, b23, c23];source canonical open family=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []];actual open index=4,U=[b23, c23]

ALL [other_index,union,intersection] rows=[[0, [a23, b23, c23], [b23, c23]], [1, [a23, b23, c23], [b23]], [2, [a23, b23, c23], [c23]], [3, [a23, b23, c23], []], [4, [b23, c23], [b23, c23]], [5, [b23, c23], [b23]], [6, [b23, c23], [c23]], [7, [b23, c23], []]];empty/full present;family qualification passed with exact source English collation

ORIGINAL carrier order=[a23, b23, c23];source canonical open family=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []];actual open index=5,U=[b23]

ALL [other_index,union,intersection] rows=[[0, [a23, b23, c23], [b23]], [1, [a23, b23], [b23]], [2, [a23, b23, c23], []], [3, [a23, b23], []], [4, [b23, c23], [b23]], [5, [b23], [b23]], [6, [b23, c23], []], [7, [b23], []]];empty/full present;family qualification passed with exact source English collation

ORIGINAL carrier order=[a23, b23, c23];source canonical open family=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []];actual open index=6,U=[c23]

ALL [other_index,union,intersection] rows=[[0, [a23, b23, c23], [c23]], [1, [a23, b23, c23], []], [2, [a23, c23], [c23]], [3, [a23, c23], []], [4, [b23, c23], [c23]], [5, [b23, c23], []], [6, [c23], [c23]], [7, [c23], []]];empty/full present;family qualification passed with exact source English collation

ORIGINAL carrier order=[a23, b23, c23];source canonical open family=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []];actual open index=7,U=[]

ALL [other_index,union,intersection] rows=[[0, [a23, b23, c23], []], [1, [a23, b23], []], [2, [a23, c23], []], [3, [a23], []], [4, [b23, c23], []], [5, [b23], []], [6, [c23], []], [7, [], []]];empty/full present;family qualification passed with exact source English collation

Hausdorff ORIGINAL designated pair:ORIGINAL distinct carrier points x=a23,y=b23;source open order=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []]

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a23, b23, c23], 否], [0, 1, [a23, b23], 否], [0, 4, [b23, c23], 否], [0, 5, [b23], 否], [1, 0, [a23, b23], 否], [1, 1, [a23, b23], 否], [1, 4, [b23], 否], [1, 5, [b23], 否], [2, 0, [a23, c23], 否], [2, 1, [a23], 否], [2, 4, [c23], 否], [2, 5, [], 是]];FIRST source-order witness=[[a23, c23], [b23]]

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

(a23, b23, [[a23, c23], [b23]], 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。