第 21679 題：緊緻子集

本頁導覽

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

R0042-21679 · q-20261014-advance-20261109-r03-compact-subset-0054

原題與已知條件

完整有限subspace topology與relative open covers。操作=compact-subcover-certificate；資料={"n":5,"parent_topology":[[],[0],[0,1],[0,1,2,3,4],[0,1,3],[0,1,3,4]],"subset":[1,2,4],"cover":[[],[1],[1,2,4],[1,4]],"selected":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

G001 — 完整有限subspace topology與relative open covers。操作=compact-subcover-certificate；資料={"n":5,"parent_topology":[[],[0],[0,1],[0,1,2,3,4],[0,1,3],[0,1,3,4]],"subset":[1,2,4],"cover":[[],[1],[1,2,4],[1,4]],"selected":[2]}。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

逐步推導

S001 · implies

操作前

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

操作後

explicit ordered source output fields=[selected_indices, selected_sets, covers_subset, selected_from_original, certificate]

條件：無

原題歷史介面標記保留，本頁直接列來源有序欄位，不需JavaScript。tag不是額外證據，固定No保留其原聲稱範圍。

步驟識別與依賴

規則：source_ordered_interface_explicit_without_javascript · 依賴：G001

S002 · implies

操作前

ORIGINAL parent Fin(5);parent open index=0,U=[];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]]

操作後

ALL [other_open_index,union,intersection] rows=[[0, [], []], [1, [0], []], [2, [0, 1], []], [3, [0, 1, 2, 3, 4], []], [4, [0, 1, 3], []], [5, [0, 1, 3, 4], []]];each belongs to the SAME parent family

條件：無

先驗母拓撲含空集與全集，再核全部有序open pair的union／intersection閉合。有限family的任意union可去重成有限union，故二元閉合足夠；不能把不合格母資料當True。

步驟識別與依賴

規則：subtop_qualified_parent_all_pairs_union_intersection · 依賴：G001, S001

S003 · implies

操作前

ORIGINAL parent Fin(5);parent open index=1,U=[0];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]]

操作後

ALL [other_open_index,union,intersection] rows=[[0, [0], []], [1, [0], [0]], [2, [0, 1], [0]], [3, [0, 1, 2, 3, 4], [0]], [4, [0, 1, 3], [0]], [5, [0, 1, 3, 4], [0]]];each belongs to the SAME parent family

條件：無

步驟識別與依賴

規則：subtop_qualified_parent_all_pairs_union_intersection · 依賴：G001, S002

S004 · implies

操作前

ORIGINAL parent Fin(5);parent open index=2,U=[0, 1];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]]

操作後

ALL [other_open_index,union,intersection] rows=[[0, [0, 1], []], [1, [0, 1], [0]], [2, [0, 1], [0, 1]], [3, [0, 1, 2, 3, 4], [0, 1]], [4, [0, 1, 3], [0, 1]], [5, [0, 1, 3, 4], [0, 1]]];each belongs to the SAME parent family

條件：無

步驟識別與依賴

規則：subtop_qualified_parent_all_pairs_union_intersection · 依賴：G001, S003

S005 · implies

操作前

ORIGINAL parent Fin(5);parent open index=3,U=[0, 1, 2, 3, 4];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]]

操作後

ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2, 3, 4], []], [1, [0, 1, 2, 3, 4], [0]], [2, [0, 1, 2, 3, 4], [0, 1]], [3, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4]], [4, [0, 1, 2, 3, 4], [0, 1, 3]], [5, [0, 1, 2, 3, 4], [0, 1, 3, 4]]];each belongs to the SAME parent family

條件：無

步驟識別與依賴

規則：subtop_qualified_parent_all_pairs_union_intersection · 依賴：G001, S004

S006 · implies

操作前

ORIGINAL parent Fin(5);parent open index=4,U=[0, 1, 3];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]]

操作後

ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 3], []], [1, [0, 1, 3], [0]], [2, [0, 1, 3], [0, 1]], [3, [0, 1, 2, 3, 4], [0, 1, 3]], [4, [0, 1, 3], [0, 1, 3]], [5, [0, 1, 3, 4], [0, 1, 3]]];each belongs to the SAME parent family

條件：無

步驟識別與依賴

規則：subtop_qualified_parent_all_pairs_union_intersection · 依賴：G001, S005

S007 · implies

操作前

ORIGINAL parent Fin(5);parent open index=5,U=[0, 1, 3, 4];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]]

操作後

ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 3, 4], []], [1, [0, 1, 3, 4], [0]], [2, [0, 1, 3, 4], [0, 1]], [3, [0, 1, 2, 3, 4], [0, 1, 3, 4]], [4, [0, 1, 3, 4], [0, 1, 3]], [5, [0, 1, 3, 4], [0, 1, 3, 4]]];each belongs to the SAME parent family

條件：無

步驟識別與依賴

規則：subtop_qualified_parent_all_pairs_union_intersection · 依賴：G001, S006

S008 · implies

操作前

ORIGINAL parent open index=0,U=[];ORIGINAL subset Y=[1, 2, 4]

操作後

actual U intersect Y=[];retain parent index even if another open has the same intersection

條件：無

對每個母open逐點交Y；相同交集只在誘導family去重，原母witness不能丟。U可含Y外的點，不強迫U subset Y。

步驟識別與依賴

規則：subtop_every_parent_open_intersection_with_original_Y · 依賴：G001, S007

S009 · implies

操作前

ORIGINAL parent open index=1,U=[0];ORIGINAL subset Y=[1, 2, 4]

操作後

actual U intersect Y=[];retain parent index even if another open has the same intersection

條件：無

對每個母open逐點交Y；相同交集只在誘導family去重，原母witness不能丟。U可含Y外的點，不強迫U subset Y。

步驟識別與依賴

規則：subtop_every_parent_open_intersection_with_original_Y · 依賴：G001, S008

S010 · implies

操作前

ORIGINAL parent open index=2,U=[0, 1];ORIGINAL subset Y=[1, 2, 4]

操作後

actual U intersect Y=[1];retain parent index even if another open has the same intersection

條件：無

對每個母open逐點交Y；相同交集只在誘導family去重，原母witness不能丟。U可含Y外的點，不強迫U subset Y。

步驟識別與依賴

規則：subtop_every_parent_open_intersection_with_original_Y · 依賴：G001, S009

S011 · implies

操作前

ORIGINAL parent open index=3,U=[0, 1, 2, 3, 4];ORIGINAL subset Y=[1, 2, 4]

操作後

actual U intersect Y=[1, 2, 4];retain parent index even if another open has the same intersection

條件：無

對每個母open逐點交Y；相同交集只在誘導family去重，原母witness不能丟。U可含Y外的點，不強迫U subset Y。

步驟識別與依賴

規則：subtop_every_parent_open_intersection_with_original_Y · 依賴：G001, S010

S012 · implies

操作前

ORIGINAL parent open index=4,U=[0, 1, 3];ORIGINAL subset Y=[1, 2, 4]

操作後

actual U intersect Y=[1];retain parent index even if another open has the same intersection

條件：無

對每個母open逐點交Y；相同交集只在誘導family去重，原母witness不能丟。U可含Y外的點，不強迫U subset Y。

步驟識別與依賴

規則：subtop_every_parent_open_intersection_with_original_Y · 依賴：G001, S011

S013 · implies

操作前

ORIGINAL parent open index=5,U=[0, 1, 3, 4];ORIGINAL subset Y=[1, 2, 4]

操作後

actual U intersect Y=[1, 4];retain parent index even if another open has the same intersection

條件：無

對每個母open逐點交Y；相同交集只在誘導family去重，原母witness不能丟。U可含Y外的點，不強迫U subset Y。

步驟識別與依賴

規則：subtop_every_parent_open_intersection_with_original_Y · 依賴：G001, S012

S014 · calculation

操作前

subtop distinct relative open count := sum(1, 1, 1, 1)

操作後

subtop distinct relative open count = 4

條件：無

依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。

步驟識別與依賴

規則：exact_rational_arithmetic · 依賴：G001, S013

S015 · implies

操作前

ORIGINAL Y=[1, 2, 4];ALL parent intersections=[[], [], [1], [1, 2, 4], [1], [1, 4]]

操作後

normalized relative topology=[[], [1], [1, 2, 4], [1, 4]];distinct count=4;empty and Y included;unions and finite intersections commute with intersect Y

條件：無

任意union與有限intersection都可把交Y提出，再由母拓撲封閉得到相對open。Y空時所有交集都是空集，誘導拓撲是只含空集的一個family，不是零個open。

步驟識別與依賴

規則：subtop_induced_axioms_and_empty_Y · 依賴：G001, S014

S016 · implies

操作前

ORIGINAL K=[1, 2, 4];COMPLETE finite relative open family=[[], [1], [1, 2, 4], [1, 4]];distinct open count=4

操作後

for ANY relative-open cover of K,retain one used original index per DISTINCT open;at most this many opens form a finite subcover;empty K covered by the empty subfamily

條件：無

這是全部cover的論證，不只一份測例。即使原cover有無窮重複索引，只有有限種不同open；每種選一個原索引即可。來源tag的finite指此distinct family，不宣稱所有indexed cover只有有限索引。

步驟識別與依賴

規則：compact_finite_DISTINCT_open_family_general_cover_argument · 依賴：G001, S015

S017 · implies

操作前

ORIGINAL indexed cover=[[], [1], [1, 2, 4], [1, 4]];relative-open family=[[], [1], [1, 2, 4], [1, 4]];K=[1, 2, 4]

操作後

ALL supplied members are relative-open;full union=[1, 2, 4];covers K=是;outer duplicate positions retained

條件：無

逐成員核relative-open與完整union；外層重複集合保不同原索引。一次cover成功不代替一般compactness證明；失敗列表仍照實給False。

步驟識別與依賴

規則：compact_actual_indexed_cover_relative_opens_and_union · 依賴：G001, S016

S018 · implies

操作前

ORIGINAL cover=[[], [1], [1, 2, 4], [1, 4]];selected sorted unique ORIGINAL indices=[2]

操作後

selected sets=[[1, 2, 4]];their union=[1, 2, 4];covers K=是;selected_from_original=是;NO minimum/minimal claim

條件：無

有效原索引只保證選自原cover，不保證已覆蓋K；必須另核union。空選擇只在空K（或相應覆蓋條件）可成功，不補入外來open也不冒稱最小子cover。

步驟識別與依賴

規則：compact_selected_ORIGINAL_indices_separate_actual_coverage · 依賴：G001, S017

逐步反向核對

S018 · 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。
S017 · 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。
S016 · 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。
S015 · 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。
S014 · premise_and_result_audit — 以精確有理數重算 subtop distinct relative open count，不以小數近似替代。
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

([2], [[1, 2, 4]], 是, 是, finite_subcover_selected_only_from_original_cover)

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: ba37afdd49fca1faa48fa16a020b02eb61cb078c83003d32f1510520c921fb4d
candidate record SHA-256: d3c52f83b42058138ba6f4f9b841d24aaa548e76af4bed2de66e289aba5f692f
semantic review receipt SHA-256: dd44555041dfbb6fe9936aa00b9a5f8c5982c2784ef61dd0cbd5d1a303a49c1a
question-first witness SHA-256: 9a6011263f6ac53b9a1fd57addf2f3e2f6088b54802c539f332d689480dc2d17
Lean: NOT_RUN · human review: NOT_PERFORMED

同源完整 JSON