R0046-23580 · q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0080
ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-nonhausdorff-boundary；資料={"points":["o4","u4"],"topology":[["o4","u4"],["u4"],[]],"compact_subset":["u4"]}。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-nonhausdorff-boundary；資料={"points":["o4","u4"],"topology":[["o4","u4"],["u4"],[]],"compact_subset":["u4"]}。請依 exact_ordered_tuple_declared_by_interface 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
source operation=compactclosed-nonhausdorff-boundary;original snapshot and mathematics retained
操作後
explicit ordered source output fields=[points, compact_subset, complement, Hausdorff, compact, closed, boundary]
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。
規則：source_ordered_interface_explicit_without_javascript · 依賴：G001
操作前
ORIGINAL carrier order=[o4, u4];ORIGINAL open-list order=[[o4, u4], [u4], []];point=o4;ALL given sets containing it have indices=[0]
操作後
candidate N(o4)=intersection of ALL those sets=[o4, u4];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。
規則：compactclosed_ALL_given_neighborhood_intersection_candidates · 依賴：G001, S001
操作前
ORIGINAL carrier order=[o4, u4];ORIGINAL open-list order=[[o4, u4], [u4], []];point=u4;ALL given sets containing it have indices=[0, 1]
操作後
candidate N(u4)=intersection of ALL those sets=[u4];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。
規則：compactclosed_ALL_given_neighborhood_intersection_candidates · 依賴：G001, S002
操作前
ORIGINAL ordered open masks=[3, 2, 0];ALL neighborhood-containment [x,y,N(y)_subset_N(x)] rows=[[o4, o4, 是], [o4, u4, 是], [u4, u4, 是]]
操作後
ALL carrier masks0..3 audited;EXACT qualifying U with every x in U having N(x) subset U=[0, 2, 3];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。
規則：compactclosed_ALL_carrier_subsets_exact_neighborhood_upsets_topology · 依賴：G001, S003
操作前
Compact-closed ORIGINAL pair index0-1:ORIGINAL distinct x=o4,y=u4;N(y)=[u4];open-list order=[[o4, u4], [u4], []]
操作後
ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [u4], 否]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[];FIRST ORIGINAL witness=null
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。
規則：compactclosed_FIRST_original_separator_with_complete_neighborhood_screen · 依賴：G001, S004
操作前
ORIGINAL points=[o4, u4];ALL original-index distinct-pair witnesses=[[o4, u4, null]];actual K=[u4]
操作後
Hausdorff=否;K compact=是 by finiteness:from ANY cover choose one member per K point;empty K needs no members;ambient complement=[o4],closed=否
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。
規則：compactclosed_ALL_pairs_Hausdorff_and_finite_subset_compact_separate · 依賴：G001, S005
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。([o4, u4], [u4], [o4], 否, 是, 否, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
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: a0ab20d2570a4c28337cc983f36cbb090121bed10e328bed5fc6723658b039cdcandidate record SHA-256: 55ec0f46812335be6c7486b1e1fcb56bfdf144041356f48d9b32f06669bd2bf5semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02Lean: NOT_RUN · human review: NOT_PERFORMED