R0046-23560 · q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0060
ambient Hausdorff與compact subset的分離－有限subcover責任。操作=compactclosed-neighborhood-subcover-certificate；資料={"points":["a9","b9","c9"],"topology":[["a9","b9","c9"],["a9","b9"],["a9","c9"],["a9"],["b9","c9"],["b9"],["c9"],[]],"compact_subset":["a9","b9"],"exterior_point":"c9"}。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":["a9","b9","c9"],"topology":[["a9","b9","c9"],["a9","b9"],["a9","c9"],["a9"],["b9","c9"],["b9"],["c9"],[]],"compact_subset":["a9","b9"],"exterior_point":"c9"}。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-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。
規則：source_ordered_interface_explicit_without_javascript · 依賴：G001
操作前
ORIGINAL carrier order=[a9, b9, c9];ORIGINAL open-list order=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []];point=a9;ALL given sets containing it have indices=[0, 1, 2, 3]
操作後
candidate N(a9)=intersection of ALL those sets=[a9];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=[a9, b9, c9];ORIGINAL open-list order=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []];point=b9;ALL given sets containing it have indices=[0, 1, 4, 5]
操作後
candidate N(b9)=intersection of ALL those sets=[b9];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 carrier order=[a9, b9, c9];ORIGINAL open-list order=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []];point=c9;ALL given sets containing it have indices=[0, 2, 4, 6]
操作後
candidate N(c9)=intersection of ALL those sets=[c9];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, S003
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=[[a9, a9, 是], [b9, b9, 是], [c9, c9, 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。
規則：compactclosed_ALL_carrier_subsets_exact_neighborhood_upsets_topology · 依賴：G001, S004
操作前
Compact-closed ORIGINAL pair index0-1:ORIGINAL distinct x=a9,y=b9;N(y)=[b9];open-list order=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []]
操作後
ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [b9], 否], [1, [b9], 否], [2, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a9, c9], 否], [1, [a9], 否], [4, [c9], 否], [5, [], 是]];FIRST ORIGINAL witness=[[a9, c9], [b9]]
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, S005
操作前
Compact-closed ORIGINAL pair index0-2:ORIGINAL distinct x=a9,y=c9;N(y)=[c9];open-list order=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []]
操作後
ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [c9], 否], [1, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a9, b9], 否], [2, [a9], 否], [4, [b9], 否], [6, [], 是]];FIRST ORIGINAL witness=[[a9, b9], [c9]]
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, S006
操作前
Compact-closed ORIGINAL pair index1-2:ORIGINAL distinct x=b9,y=c9;N(y)=[c9];open-list order=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []]
操作後
ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [c9], 否], [1, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a9, b9], 否], [2, [a9], 否], [4, [b9], 否], [6, [], 是]];FIRST ORIGINAL witness=[[a9, b9], [c9]]
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, S007
操作前
ORIGINAL points=[a9, b9, c9];ALL original-index distinct-pair witnesses=[[a9, b9, [[a9, c9], [b9]]], [a9, c9, [[a9, b9], [c9]]], [b9, c9, [[a9, b9], [c9]]]];actual K=[a9, b9]
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=[c9],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。
規則：compactclosed_ALL_pairs_Hausdorff_and_finite_subset_compact_separate · 依賴：G001, S008
操作前
CURRENT ambient Hausdorff=是;compact K=[a9, b9];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。
規則：compactclosed_GENERAL_Hausdorff_finite_subcover_intersection_proof · 依賴：G001, S009
操作前
Compact-closed ORIGINAL exterior versus K index0:ORIGINAL distinct x=c9,y=a9;N(y)=[a9];open-list order=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []]
操作後
ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [a9], 否], [2, [a9], 否], [4, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [b9, c9], 否], [1, [b9], 否], [2, [c9], 否], [3, [], 是]];FIRST ORIGINAL witness=[[b9, c9], [a9]]
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, S010
操作前
Compact-closed ORIGINAL exterior versus K index1:ORIGINAL distinct x=c9,y=b9;N(y)=[b9];open-list order=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [b9, c9], [b9], [c9], []]
操作後
ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [b9], 否], [2, [], 是]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a9, c9], 否], [1, [a9], 否], [4, [c9], 否], [5, [], 是]];FIRST ORIGINAL witness=[[a9, c9], [b9]]
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, S011
操作前
ORIGINAL exterior x=c9;sorted K=[a9, b9];ALL [y,FIRST original separator]= [[a9, [[b9, c9], [a9]]], [b9, [[a9, c9], [b9]]]]
操作後
finite intersection of U sides=[c9];union of V sides=[a9, b9];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。
規則：compactclosed_ACTUAL_cover_union_finite_U_intersection_original_order · 依賴：G001, S012
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。(c9, [[a9, [[b9, c9], [a9]]], [b9, [[a9, c9], [b9]]]], [c9], [a9, b9], 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。
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: b389c6ba483d6b3de6e489ab9ebb1dc5c8a70c6d2fbc1109a7c3e39d167ba225candidate record SHA-256: 210ff978dcec58608c4c7ba8cd371c99fcd6cb6e32ba1a9e32ecbe552588ec0fsemantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02Lean: NOT_RUN · human review: NOT_PERFORMED