第 23502 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

R0046-23502 · q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0027

原題與已知條件

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-finite-closedness-audit；資料={"points":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["b1","c1"],["b1"],["c1"],[]],"compact_subset":["a1","b1"]}。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-finite-closedness-audit；資料={"points":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["b1","c1"],["b1"],["c1"],[]],"compact_subset":["a1","b1"]}。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-finite-closedness-audit;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[compact_subset, complement, compact, closed, 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=[a1, b1, c1];ORIGINAL open-list order=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];point=a1;ALL given sets containing it have indices=[0, 1, 2, 3]

操作後

candidate N(a1)=intersection of ALL those sets=[a1];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=[a1, b1, c1];ORIGINAL open-list order=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];point=b1;ALL given sets containing it have indices=[0, 1, 4, 5]

操作後

candidate N(b1)=intersection of ALL those sets=[b1];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=[a1, b1, c1];ORIGINAL open-list order=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];point=c1;ALL given sets containing it have indices=[0, 2, 4, 6]

操作後

candidate N(c1)=intersection of ALL those sets=[c1];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=[[a1, a1, 是], [b1, b1, 是], [c1, c1, 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=a1,y=b1;N(y)=[b1];open-list order=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []]

操作後

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [b1], 否], [1, [b1], 否], [2, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a1, c1], 否], [1, [a1], 否], [4, [c1], 否], [5, [], 是]];FIRST ORIGINAL witness=[[a1, c1], [b1]]

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=a1,y=c1;N(y)=[c1];open-list order=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []]

操作後

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [c1], 否], [1, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a1, b1], 否], [2, [a1], 否], [4, [b1], 否], [6, [], 是]];FIRST ORIGINAL witness=[[a1, b1], [c1]]

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=b1,y=c1;N(y)=[c1];open-list order=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []]

操作後

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [c1], 否], [1, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a1, b1], 否], [2, [a1], 否], [4, [b1], 否], [6, [], 是]];FIRST ORIGINAL witness=[[a1, b1], [c1]]

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=[a1, b1, c1];ALL original-index distinct-pair witnesses=[[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]];actual K=[a1, b1]

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=[c1],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=[a1, b1];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

([a1, b1], [c1], 是, 是, complement_is_open_so_compact_subset_is_closed)

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: 42603182e7c2aaeb60b2779b9d75214e618f514a29dfe3c9e72bf54d2683e45d
candidate record SHA-256: 534f9226a1cca9510d4327f2f20ddf472607cc81749f65f688476de4f494300a
semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
Lean: NOT_RUN · human review: NOT_PERFORMED

同源完整 JSON