第 22967 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":["a16","b16","c16"],"topology":[["a16","b16","c16"],["a16","b16"],["a16","c16"],["a16"],["b16","c16"],["b16"],["c16"],[]],"x":"a16","y":"b16"}。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":["a16","b16","c16"],"topology":[["a16","b16","c16"],["a16","b16"],["a16","c16"],["a16"],["b16","c16"],["b16"],["c16"],[]],"x":"a16","y":"b16"}。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=[a16, b16, c16];source canonical open family=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []];actual open index=0,U=[a16, b16, c16]

ALL [other_index,union,intersection] rows=[[0, [a16, b16, c16], [a16, b16, c16]], [1, [a16, b16, c16], [a16, b16]], [2, [a16, b16, c16], [a16, c16]], [3, [a16, b16, c16], [a16]], [4, [a16, b16, c16], [b16, c16]], [5, [a16, b16, c16], [b16]], [6, [a16, b16, c16], [c16]], [7, [a16, b16, c16], []]];empty/full present;family qualification passed with exact source English collation

ORIGINAL carrier order=[a16, b16, c16];source canonical open family=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []];actual open index=1,U=[a16, b16]

ALL [other_index,union,intersection] rows=[[0, [a16, b16, c16], [a16, b16]], [1, [a16, b16], [a16, b16]], [2, [a16, b16, c16], [a16]], [3, [a16, b16], [a16]], [4, [a16, b16, c16], [b16]], [5, [a16, b16], [b16]], [6, [a16, b16, c16], []], [7, [a16, b16], []]];empty/full present;family qualification passed with exact source English collation

ORIGINAL carrier order=[a16, b16, c16];source canonical open family=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []];actual open index=2,U=[a16, c16]

ALL [other_index,union,intersection] rows=[[0, [a16, b16, c16], [a16, c16]], [1, [a16, b16, c16], [a16]], [2, [a16, c16], [a16, c16]], [3, [a16, c16], [a16]], [4, [a16, b16, c16], [c16]], [5, [a16, b16, c16], []], [6, [a16, c16], [c16]], [7, [a16, c16], []]];empty/full present;family qualification passed with exact source English collation

ORIGINAL carrier order=[a16, b16, c16];source canonical open family=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []];actual open index=3,U=[a16]

ALL [other_index,union,intersection] rows=[[0, [a16, b16, c16], [a16]], [1, [a16, b16], [a16]], [2, [a16, c16], [a16]], [3, [a16], [a16]], [4, [a16, b16, c16], []], [5, [a16, b16], []], [6, [a16, c16], []], [7, [a16], []]];empty/full present;family qualification passed with exact source English collation

ORIGINAL carrier order=[a16, b16, c16];source canonical open family=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []];actual open index=4,U=[b16, c16]

ALL [other_index,union,intersection] rows=[[0, [a16, b16, c16], [b16, c16]], [1, [a16, b16, c16], [b16]], [2, [a16, b16, c16], [c16]], [3, [a16, b16, c16], []], [4, [b16, c16], [b16, c16]], [5, [b16, c16], [b16]], [6, [b16, c16], [c16]], [7, [b16, c16], []]];empty/full present;family qualification passed with exact source English collation

ORIGINAL carrier order=[a16, b16, c16];source canonical open family=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []];actual open index=5,U=[b16]

ALL [other_index,union,intersection] rows=[[0, [a16, b16, c16], [b16]], [1, [a16, b16], [b16]], [2, [a16, b16, c16], []], [3, [a16, b16], []], [4, [b16, c16], [b16]], [5, [b16], [b16]], [6, [b16, c16], []], [7, [b16], []]];empty/full present;family qualification passed with exact source English collation

ORIGINAL carrier order=[a16, b16, c16];source canonical open family=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []];actual open index=6,U=[c16]

ALL [other_index,union,intersection] rows=[[0, [a16, b16, c16], [c16]], [1, [a16, b16, c16], []], [2, [a16, c16], [c16]], [3, [a16, c16], []], [4, [b16, c16], [c16]], [5, [b16, c16], []], [6, [c16], [c16]], [7, [c16], []]];empty/full present;family qualification passed with exact source English collation

ORIGINAL carrier order=[a16, b16, c16];source canonical open family=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []];actual open index=7,U=[]

ALL [other_index,union,intersection] rows=[[0, [a16, b16, c16], []], [1, [a16, b16], []], [2, [a16, c16], []], [3, [a16], []], [4, [b16, c16], []], [5, [b16], []], [6, [c16], []], [7, [], []]];empty/full present;family qualification passed with exact source English collation

Hausdorff ORIGINAL designated pair:ORIGINAL distinct carrier points x=a16,y=b16;source open order=[[a16, b16, c16], [a16, b16], [a16, c16], [a16], [b16, c16], [b16], [c16], []]

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a16, b16, c16], 否], [0, 1, [a16, b16], 否], [0, 4, [b16, c16], 否], [0, 5, [b16], 否], [1, 0, [a16, b16], 否], [1, 1, [a16, b16], 否], [1, 4, [b16], 否], [1, 5, [b16], 否], [2, 0, [a16, c16], 否], [2, 1, [a16], 否], [2, 4, [c16], 否], [2, 5, [], 是]];FIRST source-order witness=[[a16, c16], [b16]]

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

(a16, b16, [[a16, c16], [b16]], 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。