第 23531 題：Hausdorff空間中的緊緻集為閉集

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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

R0046-23531 · q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0006

原題與已知條件

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a5","b5","c5"],"topology":[["a5","b5","c5"],["a5","b5"],["a5","c5"],["a5"],["b5","c5"],["b5"],["c5"],[]],"compact_subset":["a5","b5"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

G001 — ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-hypotheses-qualification；資料={"points":["a5","b5","c5"],"topology":[["a5","b5","c5"],["a5","b5"],["a5","c5"],["a5"],["b5","c5"],["b5"],["c5"],[]],"compact_subset":["a5","b5"]}。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-compact-subset-closed-in-hausdorff

逐步推導

S001 · implies

操作前

source operation=compactclosed-hypotheses-qualification;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[points, compact_subset, pair_separation_rows, Hausdorff, compact, certificate]

條件：無

保留原題介面快照，本頁直接列來源欄序，不需JavaScript。radius由原mode選6／11欄，不補欄；No與實際否定依來源責任分清。

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=[a5, b5, c5];ORIGINAL open-list order=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []];point=a5;ALL given sets containing it have indices=[0, 1, 2, 3]

操作後

candidate N(a5)=intersection of ALL those sets=[a5];no original open-list reordering

條件：無

先由全部給定鄰域求交集候選，不先假設族已是拓撲；source保原open清單次序，內部元素才按UTF16排序。

步驟識別與依賴

規則：compactclosed_ALL_given_neighborhood_intersection_candidates · 依賴：G001, S001

S003 · implies

操作前

ORIGINAL carrier order=[a5, b5, c5];ORIGINAL open-list order=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []];point=b5;ALL given sets containing it have indices=[0, 1, 4, 5]

操作後

candidate N(b5)=intersection of ALL those sets=[b5];no original open-list reordering

條件：無

先由全部給定鄰域求交集候選，不先假設族已是拓撲；source保原open清單次序，內部元素才按UTF16排序。

步驟識別與依賴

規則：compactclosed_ALL_given_neighborhood_intersection_candidates · 依賴：G001, S002

S004 · implies

操作前

ORIGINAL carrier order=[a5, b5, c5];ORIGINAL open-list order=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []];point=c5;ALL given sets containing it have indices=[0, 2, 4, 6]

操作後

candidate N(c5)=intersection of ALL those sets=[c5];no original open-list reordering

條件：無

先由全部給定鄰域求交集候選，不先假設族已是拓撲；source保原open清單次序，內部元素才按UTF16排序。

步驟識別與依賴

規則：compactclosed_ALL_given_neighborhood_intersection_candidates · 依賴：G001, S003

S005 · implies

操作前

ORIGINAL ordered open masks=[7, 3, 5, 1, 6, 2, 4, 0];ALL neighborhood-containment [x,y,N(y)_subset_N(x)] rows=[[a5, a5, 是], [b5, b5, 是], [c5, c5, 是]]

操作後

ALL carrier masks0..7 audited;EXACT qualifying U with every x in U having N(x) subset U=[0, 1, 2, 3, 4, 5, 6, 7];given family equals this COMPLETE upset family

條件：無

有限拓撲的open恰是其最小鄰域的upsets；反向逐全部carrier子集合比對後，upsets在任意union及有限intersection下封閉，含空／全，故得到真正拓撲與N(x) open。這涵蓋全部open-pair責任，不只列取樣，也不必打印數百萬pairs。

步驟識別與依賴

規則：compactclosed_ALL_carrier_subsets_exact_neighborhood_upsets_topology · 依賴：G001, S004

S006 · implies

操作前

Compact-closed ORIGINAL pair index0-1:ORIGINAL distinct x=a5,y=b5;N(y)=[b5];open-list order=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []]

操作後

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [b5], 否], [1, [b5], 否], [2, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a5, c5], 否], [1, [a5], 否], [4, [c5], 否], [5, [], 是]];FIRST ORIGINAL witness=[[a5, c5], [b5]]

條件：無

若U交N(y)非空，所有含y的open都含N(y)，故不可能與U不交；若為空，N(y)本身是可用open，再按原V順序找首個。這與完整原U/V搜尋同一首見證，不是另選較小鄰域代答案。

步驟識別與依賴

規則：compactclosed_FIRST_original_separator_with_complete_neighborhood_screen · 依賴：G001, S005

S007 · implies

操作前

Compact-closed ORIGINAL pair index0-2:ORIGINAL distinct x=a5,y=c5;N(y)=[c5];open-list order=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []]

操作後

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [c5], 否], [1, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a5, b5], 否], [2, [a5], 否], [4, [b5], 否], [6, [], 是]];FIRST ORIGINAL witness=[[a5, b5], [c5]]

條件：無

步驟識別與依賴

規則：compactclosed_FIRST_original_separator_with_complete_neighborhood_screen · 依賴：G001, S006

S008 · implies

操作前

Compact-closed ORIGINAL pair index1-2:ORIGINAL distinct x=b5,y=c5;N(y)=[c5];open-list order=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []]

操作後

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [c5], 否], [1, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a5, b5], 否], [2, [a5], 否], [4, [b5], 否], [6, [], 是]];FIRST ORIGINAL witness=[[a5, b5], [c5]]

條件：無

步驟識別與依賴

規則：compactclosed_FIRST_original_separator_with_complete_neighborhood_screen · 依賴：G001, S007

S009 · implies

操作前

ORIGINAL points=[a5, b5, c5];ALL original-index distinct-pair witnesses=[[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]];actual K=[a5, b5]

操作後

Hausdorff=是;K compact=是 by finiteness:from ANY cover choose one member per K point;empty K needs no members;ambient complement=[c5],closed=是

條件：無

有限K的compact性不替代母空間Hausdorff責任；pair空域vacuous True。Closed查同一母空間的補集open，不擅自取相對補集。

步驟識別與依賴

規則：compactclosed_ALL_pairs_Hausdorff_and_finite_subset_compact_separate · 依賴：G001, S008

S010 · implies

操作前

CURRENT ambient Hausdorff=是;compact K=[a5, b5];consider ANY exterior x

操作後

for EACH y in K choose disjoint opens U_y containing x,V_y containing y;the V_y cover K;compactness picks FINITELY MANY indices;their U_y intersection is open,contains x and avoids ALL K,so X minus K is open

條件：無

一般定理靠compact選有限subcover，不是只靠此有限模型的數值。每個K點由所選某個V覆蓋，對應U與V不交；有限U交便排除全部K。空K的空交是X。此方向不主張closed implies compact。

步驟識別與依賴

規則：compactclosed_GENERAL_Hausdorff_finite_subcover_intersection_proof · 依賴：G001, S009

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

([a5, b5, c5], [a5, b5], [[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]], 是, 是, complete_finite_topology_Hausdorff_and_finite_subset_compact)

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: c5d24c7a39bbd6e05b216e0840466654cba94d4578bd8c95f227deb1b28a1e8c
candidate record SHA-256: 5a15c6f3dd1b90b21cddf4e0d4c40a5125442e834bca9d9a1b831be13f8bec0a
semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
Lean: NOT_RUN · human review: NOT_PERFORMED

同源完整 JSON