R0031-16172 · q-20261014-advance-20261101-r03-open-cover-and-subcover-0097
完整有限拓撲、目標集合與原始cover family。操作=cover-single-cover-boundary；資料={"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"selected":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。
G001 — 完整有限拓撲、目標集合與原始cover family。操作=cover-single-cover-boundary；資料={"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"selected":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。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。
操作前
same finite carrier Fin(5);complete original ordered topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]
操作後
unique empty and whole opens=是;all ordered open pairs=36;first union closure failure=null;first intersection closure failure=null
條件：無
先核子集表示、空集與全集、全部兩開集的聯／交仍開。因拓撲族本身有限，任意聯集只含有限種不同開集，反覆二元聯封閉便涵蓋它；有限交亦由二元交與全集得到。未供度量，不引入距離。
規則：cover_qualified_complete_topology_no_silent_metric · 依賴：G001
操作前
original cover family index=0;member=[0, 4];complete same topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]
操作後
original member open=是
條件：無
逐原始索引核開性，相同集合可以在不同索引重複；不能去重改題，也不能因它沒被選入就略過其非開性。
規則：cover_every_original_index_open_including_unused_members · 依賴：G001, S001
操作前
original cover family index=1;member=[0, 2, 3, 4];complete same topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]
操作後
original member open=是
條件：無
逐原始索引核開性，相同集合可以在不同索引重複；不能去重改題，也不能因它沒被選入就略過其非開性。
規則：cover_every_original_index_open_including_unused_members · 依賴：G001, S002
操作前
original cover family index=2;member=[0, 1, 2, 3, 4];complete same topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]
操作後
original member open=是
條件：無
逐原始索引核開性，相同集合可以在不同索引重複；不能去重改題，也不能因它沒被選入就略過其非開性。
規則：cover_every_original_index_open_including_unused_members · 依賴：G001, S003
操作前
same target=[0, 2, 3, 4];union of ALL original members=[0, 1, 2, 3, 4];nonopen original indices=[]
操作後
covers target=是;whole original open cover valid=是
條件：無
開性與覆蓋目標是獨立義務，聯集可超過目標；只要包含即可，不要求聯集等於目標。兩者同時成立才核可原cover。
規則：cover_openness_and_target_coverage_separate_conjunction · 依賴：G001, S004
操作前
raw selected ORIGINAL indices=[1];normalized duplicate-free indices=[1];selected original sets=[[0, 2, 3, 4]]
操作後
selected union=[0, 2, 3, 4];selected all open=是;selected covers target=是;original family qualified=是;valid selected subcover=是
條件：無
只依原索引取成員，排序索引但拒重複；空選取在空目標可合格。此協定另要求整個原族先為有效開覆蓋，所以好子族不能抹去未選中壞原成員。
規則：cover_selected_original_indices_whole_family_premise_retained · 依賴：G001, S005
操作前
this one original cover and selected indices=[1];valid subcover=是
操作後
proves compactness from this one check=否;one_finite_subcover_does_not_prove_compactness
條件：無
compact定義量化所有開覆蓋，一份有限subcover核查不足以承擔全稱責任。這個有限空間可用有限性另證compact；No不是宣告它不compact。
規則：cover_one_instance_not_all_open_covers_or_noncompactness · 依賴：G001, S006
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。([1], 是, 否, one_finite_subcover_does_not_prove_compactness)
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。
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。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 1850e64cbe27f006f2ea8be302a9804c6fe19eca197516fa72bf65086119a194candidate record SHA-256: c89dcf0e3f8c0147164463ff2b9b54504962a0f76f3115f2456503c8e432103dsemantic review receipt SHA-256: 55bb21783e2125ecf1e67e42f83fd25bec0ccd493f911f4bb937a5e9d6f9369aquestion-first witness SHA-256: 754ddf2a6039c5ef4aae78a538c6e9a1e9a58d4825aee194e869a27b8a4c95bbLean: NOT_RUN · human review: NOT_PERFORMED