# 第 23561 題：Hausdorff空間中的緊緻集為閉集

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本頁是已完成第二對話 AI 數理與中英語義覆核的教學詳解，只支持本題明示的模型與前提。沒有人工審查或新 Lean 證明，也不提升原題或 MKO 的 Canonical 真值狀態。

R0046-23561 · q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0061



## 原題與已知條件

ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a10","b10","c10"],"topology":[["a10","b10","c10"],["a10","b10"],["a10","c10"],["a10"],["b10","c10"],["b10"],["c10"],[]],"compact_subset":["a10","b10"],"exterior_point":"c10"}。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-neighborhood-subcover-certificate；資料={"points":["a10","b10","c10"],"topology":[["a10","b10","c10"],["a10","b10"],["a10","c10"],["a10"],["b10","c10"],["b10"],["c10"],[]],"compact_subset":["a10","b10"],"exterior_point":"c10"}。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](https://ocme.evemisslab.com/zh-Hant/mko/mko-compact-subset-closed-in-hausdorff/)





## 逐步推導

S001 · implies操作前

source operation=compactclosed-neighborhood-subcover-certificate;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, 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=[a10, b10, c10];ORIGINAL open-list order=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []];point=a10;ALL given sets containing it have indices=[0, 1, 2, 3]

操作後

candidate N(a10)=intersection of ALL those sets=[a10];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=[a10, b10, c10];ORIGINAL open-list order=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []];point=b10;ALL given sets containing it have indices=[0, 1, 4, 5]

操作後

candidate N(b10)=intersection of ALL those sets=[b10];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=[a10, b10, c10];ORIGINAL open-list order=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []];point=c10;ALL given sets containing it have indices=[0, 2, 4, 6]

操作後

candidate N(c10)=intersection of ALL those sets=[c10];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=[[a10, a10, 是], [b10, b10, 是], [c10, c10, 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=a10,y=b10;N(y)=[b10];open-list order=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []]

操作後

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [b10], 否], [1, [b10], 否], [2, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a10, c10], 否], [1, [a10], 否], [4, [c10], 否], [5, [], 是]];FIRST ORIGINAL witness=[[a10, c10], [b10]]

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=a10,y=c10;N(y)=[c10];open-list order=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []]

操作後

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [c10], 否], [1, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a10, b10], 否], [2, [a10], 否], [4, [b10], 否], [6, [], 是]];FIRST ORIGINAL witness=[[a10, b10], [c10]]

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, S006





S008 · implies操作前

Compact-closed ORIGINAL pair index1-2:ORIGINAL distinct x=b10,y=c10;N(y)=[c10];open-list order=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []]

操作後

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [c10], 否], [1, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a10, b10], 否], [2, [a10], 否], [4, [b10], 否], [6, [], 是]];FIRST ORIGINAL witness=[[a10, b10], [c10]]

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, S007





S009 · implies操作前

ORIGINAL points=[a10, b10, c10];ALL original-index distinct-pair witnesses=[[a10, b10, [[a10, c10], [b10]]], [a10, c10, [[a10, b10], [c10]]], [b10, c10, [[a10, b10], [c10]]]];actual K=[a10, b10]

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=[c10],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=[a10, b10];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





S011 · implies操作前

Compact-closed ORIGINAL exterior versus K index0:ORIGINAL distinct x=c10,y=a10;N(y)=[a10];open-list order=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []]

操作後

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [a10], 否], [2, [a10], 否], [4, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [b10, c10], 否], [1, [b10], 否], [2, [c10], 否], [3, [], 是]];FIRST ORIGINAL witness=[[b10, c10], [a10]]

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, S010





S012 · implies操作前

Compact-closed ORIGINAL exterior versus K index1:ORIGINAL distinct x=c10,y=b10;N(y)=[b10];open-list order=[[a10, b10, c10], [a10, b10], [a10, c10], [a10], [b10, c10], [b10], [c10], []]

操作後

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [b10], 否], [2, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a10, c10], 否], [1, [a10], 否], [4, [c10], 否], [5, [], 是]];FIRST ORIGINAL witness=[[a10, c10], [b10]]

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, S011





S013 · implies操作前

ORIGINAL exterior x=c10;sorted K=[a10, b10];ALL [y,FIRST original separator]= [[a10, [[b10, c10], [a10]]], [b10, [[a10, c10], [b10]]]]

操作後

finite intersection of U sides=[c10];union of V sides=[a10, b10];covers K=是;intersection avoids K=是;intersection open=是

條件：無

V聯集可超出K而仍是cover；U交要真open、含x、避K。沒有最小／唯一subcover聲稱；空K時交集保原RAWcarrier次序，不改成排序後列表。

步驟識別與依賴規則：compactclosed_ACTUAL_cover_union_finite_U_intersection_original_order · 依賴：G001, S012









## 逐步反向核對


- S013 · 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。
- S012 · 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。
- 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 — 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

(c10, [[a10, [[b10, c10], [a10]]], [b10, [[a10, c10], [b10]]]], [c10], [a10, b10], 是, 是, 是, finite_subcover_and_finite_intersection_separate_x_from_K)

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: 3a81b8582d311a6da6a35f9f027f0349c4b74f56d8f6684a01b27c82268c7f75
- candidate record SHA-256: e0eda35c69dc819adfb3c49a3792be691c1a23ecd23b66866a0df0c6a54bb63d
- semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
- question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
- Lean: NOT_RUN · human review: NOT_PERFORMED

[同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0061.json)