# 第 20646 題：拓撲基底

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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。

R0040-20646 · q-20261014-advance-20261107-r03-topological-basis-0071



## 原題與已知條件

完整有限carrier與完整basis candidate。操作=topbasis-existing-topology-audit；資料={"n":5,"basis":[[0],[1],[1,2],[2],[3],[3,4],[4]],"topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。


- G001 — 完整有限carrier與完整basis candidate。操作=topbasis-existing-topology-audit；資料={"n":5,"basis":[[0],[1],[1,2],[2],[3],[3,4],[4]],"topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]]}。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-topological-basis](https://ocme.evemisslab.com/zh-Hant/mko/mko-topological-basis/)





## 逐步推導

S001 · implies操作前

source operation=topbasis-existing-topology-audit;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology]

條件：無

原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。標籤不自動是數學證據，No保持來源的聲稱範圍。

步驟識別與依賴規則：source_ordered_interface_explicit_without_javascript · 依賴：G001





S002 · implies操作前

original basis candidate:ORIGINAL complete candidate family=[[0], [1], [1, 2], [2], [3], [3, 4], [4]];carrier Fin(5)

操作後

within-set numeric normalization=[[0], [1], [1, 2], [2], [3], [3, 4], [4]];no duplicate points or duplicate FAMILY elements;WHOLE original family already in required key order;empty family allowed as candidate

條件：無

basis是subset族，不是向量basis；個數可大於n。只規範每個集合內的點順序，不重排原族索引，不刪重複來修資料；空candidate族仍可受資格審查。

步驟識別與依賴規則：topbasis_canonical_candidate_family_not_semantic_reordering · 依賴：G001, S001





S003 · implies操作前

COMPLETE original candidate basis=[[0], [1], [1, 2], [2], [3], [3, 4], [4]];ALL carrier points0..4

操作後

ALL [point,containing_basis_indices] rows=[[0, [0]], [1, [1, 2]], [2, [2, 3]], [3, [4, 5]], [4, [5, 6]]];covers WHOLE carrier=是

條件：無

每個點至少落一個basis候選才cover；這項不能代替局部交細化。族為空時n≥1，所以cover失敗而不是拋棄該candidate。

步驟識別與依賴規則：topbasis_every_point_cover_separate_from_refinement · 依賴：G001, S002





S004 · implies操作前

ORIGINAL ordered basis positions(left,right)=[0, 0];actual basis elements=[[0], [0]];FULL intersection=[0]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S003





S005 · implies操作前

ORIGINAL ordered basis positions(left,right)=[0, 1];actual basis elements=[[0], [1]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S004





S006 · implies操作前

ORIGINAL ordered basis positions(left,right)=[0, 2];actual basis elements=[[0], [1, 2]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S005





S007 · implies操作前

ORIGINAL ordered basis positions(left,right)=[0, 3];actual basis elements=[[0], [2]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S006





S008 · implies操作前

ORIGINAL ordered basis positions(left,right)=[0, 4];actual basis elements=[[0], [3]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S007





S009 · implies操作前

ORIGINAL ordered basis positions(left,right)=[0, 5];actual basis elements=[[0], [3, 4]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S008





S010 · implies操作前

ORIGINAL ordered basis positions(left,right)=[0, 6];actual basis elements=[[0], [4]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S009





S011 · implies操作前

ORIGINAL ordered basis positions(left,right)=[1, 0];actual basis elements=[[1], [0]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S010





S012 · implies操作前

ORIGINAL ordered basis positions(left,right)=[1, 1];actual basis elements=[[1], [1]];FULL intersection=[1]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[1, [1]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S011





S013 · implies操作前

ORIGINAL ordered basis positions(left,right)=[1, 2];actual basis elements=[[1], [1, 2]];FULL intersection=[1]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[1, [1]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

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





S014 · implies操作前

ORIGINAL ordered basis positions(left,right)=[1, 3];actual basis elements=[[1], [2]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S013





S015 · implies操作前

ORIGINAL ordered basis positions(left,right)=[1, 4];actual basis elements=[[1], [3]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S014





S016 · implies操作前

ORIGINAL ordered basis positions(left,right)=[1, 5];actual basis elements=[[1], [3, 4]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S015





S017 · implies操作前

ORIGINAL ordered basis positions(left,right)=[1, 6];actual basis elements=[[1], [4]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S016





S018 · implies操作前

ORIGINAL ordered basis positions(left,right)=[2, 0];actual basis elements=[[1, 2], [0]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S017





S019 · implies操作前

ORIGINAL ordered basis positions(left,right)=[2, 1];actual basis elements=[[1, 2], [1]];FULL intersection=[1]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[1, [1]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S018





S020 · implies操作前

ORIGINAL ordered basis positions(left,right)=[2, 2];actual basis elements=[[1, 2], [1, 2]];FULL intersection=[1, 2]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[1, [1, 2]], [2, [2, 3]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S019





S021 · implies操作前

ORIGINAL ordered basis positions(left,right)=[2, 3];actual basis elements=[[1, 2], [2]];FULL intersection=[2]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[2, [3]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S020





S022 · implies操作前

ORIGINAL ordered basis positions(left,right)=[2, 4];actual basis elements=[[1, 2], [3]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S021





S023 · implies操作前

ORIGINAL ordered basis positions(left,right)=[2, 5];actual basis elements=[[1, 2], [3, 4]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S022





S024 · implies操作前

ORIGINAL ordered basis positions(left,right)=[2, 6];actual basis elements=[[1, 2], [4]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S023





S025 · implies操作前

ORIGINAL ordered basis positions(left,right)=[3, 0];actual basis elements=[[2], [0]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S024





S026 · implies操作前

ORIGINAL ordered basis positions(left,right)=[3, 1];actual basis elements=[[2], [1]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S025





S027 · implies操作前

ORIGINAL ordered basis positions(left,right)=[3, 2];actual basis elements=[[2], [1, 2]];FULL intersection=[2]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[2, [3]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S026





S028 · implies操作前

ORIGINAL ordered basis positions(left,right)=[3, 3];actual basis elements=[[2], [2]];FULL intersection=[2]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[2, [3]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S027





S029 · implies操作前

ORIGINAL ordered basis positions(left,right)=[3, 4];actual basis elements=[[2], [3]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S028





S030 · implies操作前

ORIGINAL ordered basis positions(left,right)=[3, 5];actual basis elements=[[2], [3, 4]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S029





S031 · implies操作前

ORIGINAL ordered basis positions(left,right)=[3, 6];actual basis elements=[[2], [4]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S030





S032 · implies操作前

ORIGINAL ordered basis positions(left,right)=[4, 0];actual basis elements=[[3], [0]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S031





S033 · implies操作前

ORIGINAL ordered basis positions(left,right)=[4, 1];actual basis elements=[[3], [1]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S032





S034 · implies操作前

ORIGINAL ordered basis positions(left,right)=[4, 2];actual basis elements=[[3], [1, 2]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S033





S035 · implies操作前

ORIGINAL ordered basis positions(left,right)=[4, 3];actual basis elements=[[3], [2]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S034





S036 · implies操作前

ORIGINAL ordered basis positions(left,right)=[4, 4];actual basis elements=[[3], [3]];FULL intersection=[3]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[3, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S035





S037 · implies操作前

ORIGINAL ordered basis positions(left,right)=[4, 5];actual basis elements=[[3], [3, 4]];FULL intersection=[3]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[3, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S036





S038 · implies操作前

ORIGINAL ordered basis positions(left,right)=[4, 6];actual basis elements=[[3], [4]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S037





S039 · implies操作前

ORIGINAL ordered basis positions(left,right)=[5, 0];actual basis elements=[[3, 4], [0]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S038





S040 · implies操作前

ORIGINAL ordered basis positions(left,right)=[5, 1];actual basis elements=[[3, 4], [1]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S039





S041 · implies操作前

ORIGINAL ordered basis positions(left,right)=[5, 2];actual basis elements=[[3, 4], [1, 2]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S040





S042 · implies操作前

ORIGINAL ordered basis positions(left,right)=[5, 3];actual basis elements=[[3, 4], [2]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S041





S043 · implies操作前

ORIGINAL ordered basis positions(left,right)=[5, 4];actual basis elements=[[3, 4], [3]];FULL intersection=[3]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[3, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S042





S044 · implies操作前

ORIGINAL ordered basis positions(left,right)=[5, 5];actual basis elements=[[3, 4], [3, 4]];FULL intersection=[3, 4]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[3, [4, 5]], [4, [5, 6]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S043





S045 · implies操作前

ORIGINAL ordered basis positions(left,right)=[5, 6];actual basis elements=[[3, 4], [4]];FULL intersection=[4]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[4, [6]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S044





S046 · implies操作前

ORIGINAL ordered basis positions(left,right)=[6, 0];actual basis elements=[[4], [0]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S045





S047 · implies操作前

ORIGINAL ordered basis positions(left,right)=[6, 1];actual basis elements=[[4], [1]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S046





S048 · implies操作前

ORIGINAL ordered basis positions(left,right)=[6, 2];actual basis elements=[[4], [1, 2]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S047





S049 · implies操作前

ORIGINAL ordered basis positions(left,right)=[6, 3];actual basis elements=[[4], [2]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S048





S050 · implies操作前

ORIGINAL ordered basis positions(left,right)=[6, 4];actual basis elements=[[4], [3]];FULL intersection=[]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S049





S051 · implies操作前

ORIGINAL ordered basis positions(left,right)=[6, 5];actual basis elements=[[4], [3, 4]];FULL intersection=[4]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[4, [6]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S050





S052 · implies操作前

ORIGINAL ordered basis positions(left,right)=[6, 6];actual basis elements=[[4], [4]];FULL intersection=[4]

操作後

ALL [intersection_point,ALL refining original basis indices] rows=[[4, [6]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

條件：無

對交集內每個x要存在某basis元素含x且包含在交集中，不要求整個交集本身在basis。逐全部候選refiner才可證不存在，首失敗保left/right/point的LEX順序。

步驟識別與依賴規則：topbasis_full_local_point_refinement_not_intersection_membership · 依賴：G001, S051





S053 · implies操作前

ACTUAL cover=是;ALL local point obligations refined=是;first failure[left,right,point,intersection]=null

操作後

abstract basis validity=是 is cover AND LOCAL refinement;not closure of the family under intersections and not a given-topology certificate yet

條件：無

兩項合取才給抽象basis；cover-only不夠，另一方面好basis不必對交運算封閉。相對已給topology還有開性與生成所有開集的其他責任。

步驟識別與依賴規則：topbasis_cover_and_local_refinement_conjunction · 依賴：G001, S052





S054 · implies操作前

given topology candidate:ORIGINAL complete candidate family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]];carrier Fin(5)

操作後

within-set numeric normalization=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]];no duplicate points or duplicate FAMILY elements;WHOLE original family already in required key order;empty family allowed as candidate

條件：無

basis是subset族，不是向量basis；個數可大於n。只規範每個集合內的點順序，不重排原族索引，不刪重複來修資料；空candidate族仍可受資格審查。

步驟識別與依賴規則：topbasis_canonical_candidate_family_not_semantic_reordering · 依賴：G001, S053





S055 · implies操作前

COMPLETE ORIGINAL given topology CANDIDATE=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]];carrier Fin(5)

操作後

empty-set listed=是;full-carrier listed=是;candidate not assumed to satisfy topology axioms

條件：無

此操作審查一個已給候選族，壞候選保False旗標，不以shape合格就宣稱topology，也不一律拋錯而漏掉題目要求的審查。

步驟識別與依賴規則：topbasis_given_topology_candidate_not_blanket_assumed_valid · 依賴：G001, S054





S056 · implies操作前

given candidate left-open position=0;set=[];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[], 0, [], 0], [[0], 1, [], 0], [[0, 1], 2, [], 0], [[0, 1, 2], 3, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 3], 7, [], 0], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 4], 9, [], 0], [[0, 2], 10, [], 0], [[0, 2, 3], 11, [], 0], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 4], 13, [], 0], [[0, 3], 14, [], 0], [[0, 3, 4], 15, [], 0], [[0, 4], 16, [], 0], [[1], 17, [], 0], [[1, 2], 18, [], 0], [[1, 2, 3], 19, [], 0], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 4], 21, [], 0], [[1, 3], 22, [], 0], [[1, 3, 4], 23, [], 0], [[1, 4], 24, [], 0], [[2], 25, [], 0], [[2, 3], 26, [], 0], [[2, 3, 4], 27, [], 0], [[2, 4], 28, [], 0], [[3], 29, [], 0], [[3, 4], 30, [], 0], [[4], 31, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S055





S057 · implies操作前

given candidate left-open position=1;set=[0];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0], 1, [], 0], [[0], 1, [0], 1], [[0, 1], 2, [0], 1], [[0, 1, 2], 3, [0], 1], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 4], 6, [0], 1], [[0, 1, 3], 7, [0], 1], [[0, 1, 3, 4], 8, [0], 1], [[0, 1, 4], 9, [0], 1], [[0, 2], 10, [0], 1], [[0, 2, 3], 11, [0], 1], [[0, 2, 3, 4], 12, [0], 1], [[0, 2, 4], 13, [0], 1], [[0, 3], 14, [0], 1], [[0, 3, 4], 15, [0], 1], [[0, 4], 16, [0], 1], [[0, 1], 2, [], 0], [[0, 1, 2], 3, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 3], 7, [], 0], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 4], 9, [], 0], [[0, 2], 10, [], 0], [[0, 2, 3], 11, [], 0], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 4], 13, [], 0], [[0, 3], 14, [], 0], [[0, 3, 4], 15, [], 0], [[0, 4], 16, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S056





S058 · implies操作前

given candidate left-open position=2;set=[0, 1];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1], 2, [], 0], [[0, 1], 2, [0], 1], [[0, 1], 2, [0, 1], 2], [[0, 1, 2], 3, [0, 1], 2], [[0, 1, 2, 3], 4, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 2, 4], 6, [0, 1], 2], [[0, 1, 3], 7, [0, 1], 2], [[0, 1, 3, 4], 8, [0, 1], 2], [[0, 1, 4], 9, [0, 1], 2], [[0, 1, 2], 3, [0], 1], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 4], 6, [0], 1], [[0, 1, 3], 7, [0], 1], [[0, 1, 3, 4], 8, [0], 1], [[0, 1, 4], 9, [0], 1], [[0, 1], 2, [1], 17], [[0, 1, 2], 3, [1], 17], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 3], 7, [1], 17], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 4], 9, [1], 17], [[0, 1, 2], 3, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 3], 7, [], 0], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 4], 9, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S057





S059 · implies操作前

given candidate left-open position=3;set=[0, 1, 2];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1, 2], 3, [], 0], [[0, 1, 2], 3, [0], 1], [[0, 1, 2], 3, [0, 1], 2], [[0, 1, 2], 3, [0, 1, 2], 3], [[0, 1, 2, 3], 4, [0, 1, 2], 3], [[0, 1, 2, 3, 4], 5, [0, 1, 2], 3], [[0, 1, 2, 4], 6, [0, 1, 2], 3], [[0, 1, 2, 3], 4, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 2, 4], 6, [0, 1], 2], [[0, 1, 2], 3, [0, 2], 10], [[0, 1, 2, 3], 4, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 4], 6, [0, 2], 10], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 4], 6, [0], 1], [[0, 1, 2], 3, [1], 17], [[0, 1, 2], 3, [1, 2], 18], [[0, 1, 2, 3], 4, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 4], 6, [1, 2], 18], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 2], 3, [2], 25], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S058





S060 · implies操作前

given candidate left-open position=4;set=[0, 1, 2, 3];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3], 4, [0, 1], 2], [[0, 1, 2, 3], 4, [0, 1, 2], 3], [[0, 1, 2, 3], 4, [0, 1, 2, 3], 4], [[0, 1, 2, 3, 4], 5, [0, 1, 2, 3], 4], [[0, 1, 2, 3, 4], 5, [0, 1, 2], 3], [[0, 1, 2, 3], 4, [0, 1, 3], 7], [[0, 1, 2, 3, 4], 5, [0, 1, 3], 7], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 2, 3], 4, [0, 2], 10], [[0, 1, 2, 3], 4, [0, 2, 3], 11], [[0, 1, 2, 3, 4], 5, [0, 2, 3], 11], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 3], 4, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3], 4, [1, 2], 18], [[0, 1, 2, 3], 4, [1, 2, 3], 19], [[0, 1, 2, 3, 4], 5, [1, 2, 3], 19], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 3], 4, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3], 4, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S059





S061 · implies操作前

given candidate left-open position=5;set=[0, 1, 2, 3, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1, 2], 3], [[0, 1, 2, 3, 4], 5, [0, 1, 2, 3], 4], [[0, 1, 2, 3, 4], 5, [0, 1, 2, 3, 4], 5], [[0, 1, 2, 3, 4], 5, [0, 1, 2, 4], 6], [[0, 1, 2, 3, 4], 5, [0, 1, 3], 7], [[0, 1, 2, 3, 4], 5, [0, 1, 3, 4], 8], [[0, 1, 2, 3, 4], 5, [0, 1, 4], 9], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2, 3], 11], [[0, 1, 2, 3, 4], 5, [0, 2, 3, 4], 12], [[0, 1, 2, 3, 4], 5, [0, 2, 4], 13], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3, 4], 15], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2, 3], 19], [[0, 1, 2, 3, 4], 5, [1, 2, 3, 4], 20], [[0, 1, 2, 3, 4], 5, [1, 2, 4], 21], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3, 4], 23], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3, 4], 27], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S060





S062 · implies操作前

given candidate left-open position=6;set=[0, 1, 2, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 4], 6, [0], 1], [[0, 1, 2, 4], 6, [0, 1], 2], [[0, 1, 2, 4], 6, [0, 1, 2], 3], [[0, 1, 2, 3, 4], 5, [0, 1, 2], 3], [[0, 1, 2, 3, 4], 5, [0, 1, 2, 4], 6], [[0, 1, 2, 4], 6, [0, 1, 2, 4], 6], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1, 4], 9], [[0, 1, 2, 4], 6, [0, 1, 4], 9], [[0, 1, 2, 4], 6, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2, 4], 13], [[0, 1, 2, 4], 6, [0, 2, 4], 13], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 1, 2, 4], 6, [0, 4], 16], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 2, 4], 6, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2, 4], 21], [[0, 1, 2, 4], 6, [1, 2, 4], 21], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 2, 4], 6, [1, 4], 24], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 4], 6, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S061





S063 · implies操作前

given candidate left-open position=7;set=[0, 1, 3];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1, 3], 7, [], 0], [[0, 1, 3], 7, [0], 1], [[0, 1, 3], 7, [0, 1], 2], [[0, 1, 2, 3], 4, [0, 1], 2], [[0, 1, 2, 3], 4, [0, 1, 3], 7], [[0, 1, 2, 3, 4], 5, [0, 1, 3], 7], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 3], 7, [0, 1, 3], 7], [[0, 1, 3, 4], 8, [0, 1, 3], 7], [[0, 1, 3, 4], 8, [0, 1], 2], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3], 4, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 3], 7, [0, 3], 14], [[0, 1, 3, 4], 8, [0, 3], 14], [[0, 1, 3, 4], 8, [0], 1], [[0, 1, 3], 7, [1], 17], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3], 4, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 3], 7, [1, 3], 22], [[0, 1, 3, 4], 8, [1, 3], 22], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 3], 7, [3], 29], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S062





S064 · implies操作前

given candidate left-open position=8;set=[0, 1, 3, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1, 3, 4], 8, [], 0], [[0, 1, 3, 4], 8, [0], 1], [[0, 1, 3, 4], 8, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1, 3], 7], [[0, 1, 2, 3, 4], 5, [0, 1, 3, 4], 8], [[0, 1, 2, 3, 4], 5, [0, 1, 4], 9], [[0, 1, 3, 4], 8, [0, 1, 3], 7], [[0, 1, 3, 4], 8, [0, 1, 3, 4], 8], [[0, 1, 3, 4], 8, [0, 1, 4], 9], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3, 4], 15], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 1, 3, 4], 8, [0, 3], 14], [[0, 1, 3, 4], 8, [0, 3, 4], 15], [[0, 1, 3, 4], 8, [0, 4], 16], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3, 4], 23], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 3, 4], 8, [1, 3], 22], [[0, 1, 3, 4], 8, [1, 3, 4], 23], [[0, 1, 3, 4], 8, [1, 4], 24], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [3, 4], 30], [[0, 1, 3, 4], 8, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S063





S065 · implies操作前

given candidate left-open position=9;set=[0, 1, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1, 4], 9, [], 0], [[0, 1, 4], 9, [0], 1], [[0, 1, 4], 9, [0, 1], 2], [[0, 1, 2, 4], 6, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1, 4], 9], [[0, 1, 2, 4], 6, [0, 1, 4], 9], [[0, 1, 3, 4], 8, [0, 1], 2], [[0, 1, 3, 4], 8, [0, 1, 4], 9], [[0, 1, 4], 9, [0, 1, 4], 9], [[0, 1, 2, 4], 6, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 1, 2, 4], 6, [0, 4], 16], [[0, 1, 3, 4], 8, [0], 1], [[0, 1, 3, 4], 8, [0, 4], 16], [[0, 1, 4], 9, [0, 4], 16], [[0, 1, 4], 9, [1], 17], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 2, 4], 6, [1, 4], 24], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 3, 4], 8, [1, 4], 24], [[0, 1, 4], 9, [1, 4], 24], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 3, 4], 8, [4], 31], [[0, 1, 4], 9, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S064





S066 · implies操作前

given candidate left-open position=10;set=[0, 2];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 2], 10, [], 0], [[0, 2], 10, [0], 1], [[0, 1, 2], 3, [0], 1], [[0, 1, 2], 3, [0, 2], 10], [[0, 1, 2, 3], 4, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 4], 6, [0, 2], 10], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 4], 6, [0], 1], [[0, 2], 10, [0, 2], 10], [[0, 2, 3], 11, [0, 2], 10], [[0, 2, 3, 4], 12, [0, 2], 10], [[0, 2, 4], 13, [0, 2], 10], [[0, 2, 3], 11, [0], 1], [[0, 2, 3, 4], 12, [0], 1], [[0, 2, 4], 13, [0], 1], [[0, 1, 2], 3, [], 0], [[0, 1, 2], 3, [2], 25], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 2], 10, [2], 25], [[0, 2, 3], 11, [2], 25], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 4], 13, [2], 25], [[0, 2, 3], 11, [], 0], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 4], 13, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S065





S067 · implies操作前

given candidate left-open position=11;set=[0, 2, 3];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 2, 3], 11, [], 0], [[0, 2, 3], 11, [0], 1], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3], 4, [0, 2], 10], [[0, 1, 2, 3], 4, [0, 2, 3], 11], [[0, 1, 2, 3, 4], 5, [0, 2, 3], 11], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 3], 4, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 2, 3], 11, [0, 2], 10], [[0, 2, 3], 11, [0, 2, 3], 11], [[0, 2, 3, 4], 12, [0, 2, 3], 11], [[0, 2, 3, 4], 12, [0, 2], 10], [[0, 2, 3], 11, [0, 3], 14], [[0, 2, 3, 4], 12, [0, 3], 14], [[0, 2, 3, 4], 12, [0], 1], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3], 4, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 2, 3], 11, [2], 25], [[0, 2, 3], 11, [2, 3], 26], [[0, 2, 3, 4], 12, [2, 3], 26], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 3], 11, [3], 29], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S066





S068 · implies操作前

given candidate left-open position=12;set=[0, 2, 3, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 2, 3, 4], 12, [], 0], [[0, 2, 3, 4], 12, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2, 3], 11], [[0, 1, 2, 3, 4], 5, [0, 2, 3, 4], 12], [[0, 1, 2, 3, 4], 5, [0, 2, 4], 13], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3, 4], 15], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 2, 3, 4], 12, [0, 2], 10], [[0, 2, 3, 4], 12, [0, 2, 3], 11], [[0, 2, 3, 4], 12, [0, 2, 3, 4], 12], [[0, 2, 3, 4], 12, [0, 2, 4], 13], [[0, 2, 3, 4], 12, [0, 3], 14], [[0, 2, 3, 4], 12, [0, 3, 4], 15], [[0, 2, 3, 4], 12, [0, 4], 16], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3, 4], 27], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 3, 4], 12, [2, 3], 26], [[0, 2, 3, 4], 12, [2, 3, 4], 27], [[0, 2, 3, 4], 12, [2, 4], 28], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [3, 4], 30], [[0, 2, 3, 4], 12, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S067





S069 · implies操作前

given candidate left-open position=13;set=[0, 2, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 2, 4], 13, [], 0], [[0, 2, 4], 13, [0], 1], [[0, 1, 2, 4], 6, [0], 1], [[0, 1, 2, 4], 6, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2, 4], 13], [[0, 1, 2, 4], 6, [0, 2, 4], 13], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 1, 2, 4], 6, [0, 4], 16], [[0, 2, 4], 13, [0, 2], 10], [[0, 2, 3, 4], 12, [0, 2], 10], [[0, 2, 3, 4], 12, [0, 2, 4], 13], [[0, 2, 4], 13, [0, 2, 4], 13], [[0, 2, 3, 4], 12, [0], 1], [[0, 2, 3, 4], 12, [0, 4], 16], [[0, 2, 4], 13, [0, 4], 16], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 4], 6, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31], [[0, 2, 4], 13, [2], 25], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 3, 4], 12, [2, 4], 28], [[0, 2, 4], 13, [2, 4], 28], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 3, 4], 12, [4], 31], [[0, 2, 4], 13, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S068





S070 · implies操作前

given candidate left-open position=14;set=[0, 3];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 3], 14, [], 0], [[0, 3], 14, [0], 1], [[0, 1, 3], 7, [0], 1], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3], 4, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 3], 7, [0, 3], 14], [[0, 1, 3, 4], 8, [0, 3], 14], [[0, 1, 3, 4], 8, [0], 1], [[0, 2, 3], 11, [0], 1], [[0, 2, 3], 11, [0, 3], 14], [[0, 2, 3, 4], 12, [0, 3], 14], [[0, 2, 3, 4], 12, [0], 1], [[0, 3], 14, [0, 3], 14], [[0, 3, 4], 15, [0, 3], 14], [[0, 3, 4], 15, [0], 1], [[0, 1, 3], 7, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 3], 7, [3], 29], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [], 0], [[0, 2, 3], 11, [], 0], [[0, 2, 3], 11, [3], 29], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [], 0], [[0, 3], 14, [3], 29], [[0, 3, 4], 15, [3], 29], [[0, 3, 4], 15, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S069





S071 · implies操作前

given candidate left-open position=15;set=[0, 3, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 3, 4], 15, [], 0], [[0, 3, 4], 15, [0], 1], [[0, 1, 3, 4], 8, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3, 4], 15], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 1, 3, 4], 8, [0, 3], 14], [[0, 1, 3, 4], 8, [0, 3, 4], 15], [[0, 1, 3, 4], 8, [0, 4], 16], [[0, 2, 3, 4], 12, [0], 1], [[0, 2, 3, 4], 12, [0, 3], 14], [[0, 2, 3, 4], 12, [0, 3, 4], 15], [[0, 2, 3, 4], 12, [0, 4], 16], [[0, 3, 4], 15, [0, 3], 14], [[0, 3, 4], 15, [0, 3, 4], 15], [[0, 3, 4], 15, [0, 4], 16], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [3, 4], 30], [[0, 1, 3, 4], 8, [4], 31], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [3, 4], 30], [[0, 2, 3, 4], 12, [4], 31], [[0, 3, 4], 15, [3], 29], [[0, 3, 4], 15, [3, 4], 30], [[0, 3, 4], 15, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S070





S072 · implies操作前

given candidate left-open position=16;set=[0, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 4], 16, [], 0], [[0, 4], 16, [0], 1], [[0, 1, 4], 9, [0], 1], [[0, 1, 2, 4], 6, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 1, 2, 4], 6, [0, 4], 16], [[0, 1, 3, 4], 8, [0], 1], [[0, 1, 3, 4], 8, [0, 4], 16], [[0, 1, 4], 9, [0, 4], 16], [[0, 2, 4], 13, [0], 1], [[0, 2, 3, 4], 12, [0], 1], [[0, 2, 3, 4], 12, [0, 4], 16], [[0, 2, 4], 13, [0, 4], 16], [[0, 3, 4], 15, [0], 1], [[0, 3, 4], 15, [0, 4], 16], [[0, 4], 16, [0, 4], 16], [[0, 1, 4], 9, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 3, 4], 8, [4], 31], [[0, 1, 4], 9, [4], 31], [[0, 2, 4], 13, [], 0], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 3, 4], 12, [4], 31], [[0, 2, 4], 13, [4], 31], [[0, 3, 4], 15, [], 0], [[0, 3, 4], 15, [4], 31], [[0, 4], 16, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S071





S073 · implies操作前

given candidate left-open position=17;set=[1];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1], 17, [], 0], [[0, 1], 2, [], 0], [[0, 1], 2, [1], 17], [[0, 1, 2], 3, [1], 17], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 3], 7, [1], 17], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 4], 9, [1], 17], [[0, 1, 2], 3, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 3], 7, [], 0], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 4], 9, [], 0], [[1], 17, [1], 17], [[1, 2], 18, [1], 17], [[1, 2, 3], 19, [1], 17], [[1, 2, 3, 4], 20, [1], 17], [[1, 2, 4], 21, [1], 17], [[1, 3], 22, [1], 17], [[1, 3, 4], 23, [1], 17], [[1, 4], 24, [1], 17], [[1, 2], 18, [], 0], [[1, 2, 3], 19, [], 0], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 4], 21, [], 0], [[1, 3], 22, [], 0], [[1, 3, 4], 23, [], 0], [[1, 4], 24, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S072





S074 · implies操作前

given candidate left-open position=18;set=[1, 2];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1, 2], 18, [], 0], [[0, 1, 2], 3, [], 0], [[0, 1, 2], 3, [1], 17], [[0, 1, 2], 3, [1, 2], 18], [[0, 1, 2, 3], 4, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 4], 6, [1, 2], 18], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 2], 3, [2], 25], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0], [[1, 2], 18, [1], 17], [[1, 2], 18, [1, 2], 18], [[1, 2, 3], 19, [1, 2], 18], [[1, 2, 3, 4], 20, [1, 2], 18], [[1, 2, 4], 21, [1, 2], 18], [[1, 2, 3], 19, [1], 17], [[1, 2, 3, 4], 20, [1], 17], [[1, 2, 4], 21, [1], 17], [[1, 2], 18, [2], 25], [[1, 2, 3], 19, [2], 25], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 4], 21, [2], 25], [[1, 2, 3], 19, [], 0], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 4], 21, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S073





S075 · implies操作前

given candidate left-open position=19;set=[1, 2, 3];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1, 2, 3], 19, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3], 4, [1, 2], 18], [[0, 1, 2, 3], 4, [1, 2, 3], 19], [[0, 1, 2, 3, 4], 5, [1, 2, 3], 19], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 3], 4, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3], 4, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0], [[1, 2, 3], 19, [1], 17], [[1, 2, 3], 19, [1, 2], 18], [[1, 2, 3], 19, [1, 2, 3], 19], [[1, 2, 3, 4], 20, [1, 2, 3], 19], [[1, 2, 3, 4], 20, [1, 2], 18], [[1, 2, 3], 19, [1, 3], 22], [[1, 2, 3, 4], 20, [1, 3], 22], [[1, 2, 3, 4], 20, [1], 17], [[1, 2, 3], 19, [2], 25], [[1, 2, 3], 19, [2, 3], 26], [[1, 2, 3, 4], 20, [2, 3], 26], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 3], 19, [3], 29], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S074





S076 · implies操作前

given candidate left-open position=20;set=[1, 2, 3, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1, 2, 3, 4], 20, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2, 3], 19], [[0, 1, 2, 3, 4], 5, [1, 2, 3, 4], 20], [[0, 1, 2, 3, 4], 5, [1, 2, 4], 21], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3, 4], 23], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3, 4], 27], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31], [[1, 2, 3, 4], 20, [1], 17], [[1, 2, 3, 4], 20, [1, 2], 18], [[1, 2, 3, 4], 20, [1, 2, 3], 19], [[1, 2, 3, 4], 20, [1, 2, 3, 4], 20], [[1, 2, 3, 4], 20, [1, 2, 4], 21], [[1, 2, 3, 4], 20, [1, 3], 22], [[1, 2, 3, 4], 20, [1, 3, 4], 23], [[1, 2, 3, 4], 20, [1, 4], 24], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 3, 4], 20, [2, 3], 26], [[1, 2, 3, 4], 20, [2, 3, 4], 27], [[1, 2, 3, 4], 20, [2, 4], 28], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [3, 4], 30], [[1, 2, 3, 4], 20, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S075





S077 · implies操作前

given candidate left-open position=21;set=[1, 2, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1, 2, 4], 21, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 2, 4], 6, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2, 4], 21], [[0, 1, 2, 4], 6, [1, 2, 4], 21], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 2, 4], 6, [1, 4], 24], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 4], 6, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31], [[1, 2, 4], 21, [1], 17], [[1, 2, 4], 21, [1, 2], 18], [[1, 2, 3, 4], 20, [1, 2], 18], [[1, 2, 3, 4], 20, [1, 2, 4], 21], [[1, 2, 4], 21, [1, 2, 4], 21], [[1, 2, 3, 4], 20, [1], 17], [[1, 2, 3, 4], 20, [1, 4], 24], [[1, 2, 4], 21, [1, 4], 24], [[1, 2, 4], 21, [2], 25], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 3, 4], 20, [2, 4], 28], [[1, 2, 4], 21, [2, 4], 28], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 3, 4], 20, [4], 31], [[1, 2, 4], 21, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S076





S078 · implies操作前

given candidate left-open position=22;set=[1, 3];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1, 3], 22, [], 0], [[0, 1, 3], 7, [], 0], [[0, 1, 3], 7, [1], 17], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3], 4, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 3], 7, [1, 3], 22], [[0, 1, 3, 4], 8, [1, 3], 22], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 3], 7, [3], 29], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [], 0], [[1, 3], 22, [1], 17], [[1, 2, 3], 19, [1], 17], [[1, 2, 3], 19, [1, 3], 22], [[1, 2, 3, 4], 20, [1, 3], 22], [[1, 2, 3, 4], 20, [1], 17], [[1, 3], 22, [1, 3], 22], [[1, 3, 4], 23, [1, 3], 22], [[1, 3, 4], 23, [1], 17], [[1, 2, 3], 19, [], 0], [[1, 2, 3], 19, [3], 29], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [], 0], [[1, 3], 22, [3], 29], [[1, 3, 4], 23, [3], 29], [[1, 3, 4], 23, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S077





S079 · implies操作前

given candidate left-open position=23;set=[1, 3, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1, 3, 4], 23, [], 0], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3, 4], 23], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 3, 4], 8, [1, 3], 22], [[0, 1, 3, 4], 8, [1, 3, 4], 23], [[0, 1, 3, 4], 8, [1, 4], 24], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [3, 4], 30], [[0, 1, 3, 4], 8, [4], 31], [[1, 3, 4], 23, [1], 17], [[1, 2, 3, 4], 20, [1], 17], [[1, 2, 3, 4], 20, [1, 3], 22], [[1, 2, 3, 4], 20, [1, 3, 4], 23], [[1, 2, 3, 4], 20, [1, 4], 24], [[1, 3, 4], 23, [1, 3], 22], [[1, 3, 4], 23, [1, 3, 4], 23], [[1, 3, 4], 23, [1, 4], 24], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [3, 4], 30], [[1, 2, 3, 4], 20, [4], 31], [[1, 3, 4], 23, [3], 29], [[1, 3, 4], 23, [3, 4], 30], [[1, 3, 4], 23, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S078





S080 · implies操作前

given candidate left-open position=24;set=[1, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1, 4], 24, [], 0], [[0, 1, 4], 9, [], 0], [[0, 1, 4], 9, [1], 17], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 2, 4], 6, [1, 4], 24], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 3, 4], 8, [1, 4], 24], [[0, 1, 4], 9, [1, 4], 24], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 3, 4], 8, [4], 31], [[0, 1, 4], 9, [4], 31], [[1, 4], 24, [1], 17], [[1, 2, 4], 21, [1], 17], [[1, 2, 3, 4], 20, [1], 17], [[1, 2, 3, 4], 20, [1, 4], 24], [[1, 2, 4], 21, [1, 4], 24], [[1, 3, 4], 23, [1], 17], [[1, 3, 4], 23, [1, 4], 24], [[1, 4], 24, [1, 4], 24], [[1, 2, 4], 21, [], 0], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 3, 4], 20, [4], 31], [[1, 2, 4], 21, [4], 31], [[1, 3, 4], 23, [], 0], [[1, 3, 4], 23, [4], 31], [[1, 4], 24, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S079





S081 · implies操作前

given candidate left-open position=25;set=[2];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[2], 25, [], 0], [[0, 2], 10, [], 0], [[0, 1, 2], 3, [], 0], [[0, 1, 2], 3, [2], 25], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 2], 10, [2], 25], [[0, 2, 3], 11, [2], 25], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 4], 13, [2], 25], [[0, 2, 3], 11, [], 0], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 4], 13, [], 0], [[1, 2], 18, [], 0], [[1, 2], 18, [2], 25], [[1, 2, 3], 19, [2], 25], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 4], 21, [2], 25], [[1, 2, 3], 19, [], 0], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 4], 21, [], 0], [[2], 25, [2], 25], [[2, 3], 26, [2], 25], [[2, 3, 4], 27, [2], 25], [[2, 4], 28, [2], 25], [[2, 3], 26, [], 0], [[2, 3, 4], 27, [], 0], [[2, 4], 28, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S080





S082 · implies操作前

given candidate left-open position=26;set=[2, 3];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[2, 3], 26, [], 0], [[0, 2, 3], 11, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3], 4, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 2, 3], 11, [2], 25], [[0, 2, 3], 11, [2, 3], 26], [[0, 2, 3, 4], 12, [2, 3], 26], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 3], 11, [3], 29], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [], 0], [[1, 2, 3], 19, [], 0], [[1, 2, 3], 19, [2], 25], [[1, 2, 3], 19, [2, 3], 26], [[1, 2, 3, 4], 20, [2, 3], 26], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 3], 19, [3], 29], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [], 0], [[2, 3], 26, [2], 25], [[2, 3], 26, [2, 3], 26], [[2, 3, 4], 27, [2, 3], 26], [[2, 3, 4], 27, [2], 25], [[2, 3], 26, [3], 29], [[2, 3, 4], 27, [3], 29], [[2, 3, 4], 27, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S081





S083 · implies操作前

given candidate left-open position=27;set=[2, 3, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[2, 3, 4], 27, [], 0], [[0, 2, 3, 4], 12, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3, 4], 27], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 3, 4], 12, [2, 3], 26], [[0, 2, 3, 4], 12, [2, 3, 4], 27], [[0, 2, 3, 4], 12, [2, 4], 28], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [3, 4], 30], [[0, 2, 3, 4], 12, [4], 31], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 3, 4], 20, [2, 3], 26], [[1, 2, 3, 4], 20, [2, 3, 4], 27], [[1, 2, 3, 4], 20, [2, 4], 28], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [3, 4], 30], [[1, 2, 3, 4], 20, [4], 31], [[2, 3, 4], 27, [2], 25], [[2, 3, 4], 27, [2, 3], 26], [[2, 3, 4], 27, [2, 3, 4], 27], [[2, 3, 4], 27, [2, 4], 28], [[2, 3, 4], 27, [3], 29], [[2, 3, 4], 27, [3, 4], 30], [[2, 3, 4], 27, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S082





S084 · implies操作前

given candidate left-open position=28;set=[2, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[2, 4], 28, [], 0], [[0, 2, 4], 13, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 4], 6, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31], [[0, 2, 4], 13, [2], 25], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 3, 4], 12, [2, 4], 28], [[0, 2, 4], 13, [2, 4], 28], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 3, 4], 12, [4], 31], [[0, 2, 4], 13, [4], 31], [[1, 2, 4], 21, [], 0], [[1, 2, 4], 21, [2], 25], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 3, 4], 20, [2, 4], 28], [[1, 2, 4], 21, [2, 4], 28], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 3, 4], 20, [4], 31], [[1, 2, 4], 21, [4], 31], [[2, 4], 28, [2], 25], [[2, 3, 4], 27, [2], 25], [[2, 3, 4], 27, [2, 4], 28], [[2, 4], 28, [2, 4], 28], [[2, 3, 4], 27, [], 0], [[2, 3, 4], 27, [4], 31], [[2, 4], 28, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S083





S085 · implies操作前

given candidate left-open position=29;set=[3];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[3], 29, [], 0], [[0, 3], 14, [], 0], [[0, 1, 3], 7, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 3], 7, [3], 29], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [], 0], [[0, 2, 3], 11, [], 0], [[0, 2, 3], 11, [3], 29], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [], 0], [[0, 3], 14, [3], 29], [[0, 3, 4], 15, [3], 29], [[0, 3, 4], 15, [], 0], [[1, 3], 22, [], 0], [[1, 2, 3], 19, [], 0], [[1, 2, 3], 19, [3], 29], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [], 0], [[1, 3], 22, [3], 29], [[1, 3, 4], 23, [3], 29], [[1, 3, 4], 23, [], 0], [[2, 3], 26, [], 0], [[2, 3], 26, [3], 29], [[2, 3, 4], 27, [3], 29], [[2, 3, 4], 27, [], 0], [[3], 29, [3], 29], [[3, 4], 30, [3], 29], [[3, 4], 30, [], 0]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S084





S086 · implies操作前

given candidate left-open position=30;set=[3, 4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[3, 4], 30, [], 0], [[0, 3, 4], 15, [], 0], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [3, 4], 30], [[0, 1, 3, 4], 8, [4], 31], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [3, 4], 30], [[0, 2, 3, 4], 12, [4], 31], [[0, 3, 4], 15, [3], 29], [[0, 3, 4], 15, [3, 4], 30], [[0, 3, 4], 15, [4], 31], [[1, 3, 4], 23, [], 0], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [3, 4], 30], [[1, 2, 3, 4], 20, [4], 31], [[1, 3, 4], 23, [3], 29], [[1, 3, 4], 23, [3, 4], 30], [[1, 3, 4], 23, [4], 31], [[2, 3, 4], 27, [], 0], [[2, 3, 4], 27, [3], 29], [[2, 3, 4], 27, [3, 4], 30], [[2, 3, 4], 27, [4], 31], [[3, 4], 30, [3], 29], [[3, 4], 30, [3, 4], 30], [[3, 4], 30, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S085





S087 · implies操作前

given candidate left-open position=31;set=[4];ALL right candidate opens in original family order

操作後

ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[4], 31, [], 0], [[0, 4], 16, [], 0], [[0, 1, 4], 9, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 3, 4], 8, [4], 31], [[0, 1, 4], 9, [4], 31], [[0, 2, 4], 13, [], 0], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 3, 4], 12, [4], 31], [[0, 2, 4], 13, [4], 31], [[0, 3, 4], 15, [], 0], [[0, 3, 4], 15, [4], 31], [[0, 4], 16, [4], 31], [[1, 4], 24, [], 0], [[1, 2, 4], 21, [], 0], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 3, 4], 20, [4], 31], [[1, 2, 4], 21, [4], 31], [[1, 3, 4], 23, [], 0], [[1, 3, 4], 23, [4], 31], [[1, 4], 24, [4], 31], [[2, 4], 28, [], 0], [[2, 3, 4], 27, [], 0], [[2, 3, 4], 27, [4], 31], [[2, 4], 28, [4], 31], [[3, 4], 30, [], 0], [[3, 4], 30, [4], 31], [[4], 31, [4], 31]];null is an actual missing required set

條件：無

逐完整表二元union／intersection，不抽樣；缺的集合直接列出，不能替原表補集合來通過。

步驟識別與依賴規則：topbasis_given_topology_all_binary_axioms_actual_missing_sets · 依賴：G001, S086





S088 · implies操作前

given candidate empty/full flags=[是, 是];ALL binary closure flags=[是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]

操作後

topology_valid=是;IF valid,arbitrary indexed unions reduce to finitely many distinct opens,so binary union closure suffices;empty intersections use full carrier

條件：無

有限族只有有限個不同開集，故任意帶索引union等於有限union；只有空集／全集與閉包責任都通過才啟用這個topology證書。False時實際缺項仍是候選的失敗，不修改資料。

步驟識別與依賴規則：topbasis_given_topology_full_axiom_conjunction_not_only_binary_slogan · 依賴：G001, S087





S089 · implies操作前

ORIGINAL complete candidate basis in fixed family-index order=[[0], [1], [1, 2], [2], [3], [3, 4], [4]];carrier Fin(5)

操作後

prefix0 has EXACTLY the empty-subfamily union:empty set,with empty index witness;this mathematical union exists even when basis axioms fail

條件：無

空subfamily的union固定為空集合，毋須空集合本身是basis元素。此步只談union構造，不提前把不合格basis的unions稱topology。

步驟識別與依賴規則：topbasis_empty_union_initial_state_not_missing_empty_set · 依賴：G001, S088





S090 · implies操作前

next ORIGINAL basis index=0;basis set=[0];ALL previous-prefix unions in numeric POINT-mask order=[[]]

操作後

ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [0], [0]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0]]

條件：無

任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。

步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S089





S091 · implies操作前

next ORIGINAL basis index=1;basis set=[1];ALL previous-prefix unions in numeric POINT-mask order=[[], [0]]

操作後

ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [1], [1]], [[0], [0, 1], [0, 1]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [1]]

條件：無

任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。

步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S090





S092 · implies操作前

next ORIGINAL basis index=2;basis set=[1, 2];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1]]

操作後

ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [1, 2], [2]], [[0], [0, 1, 2], [0, 2]], [[1], [1, 2], [1, 2]], [[0, 1], [0, 1, 2], [0, 1, 2]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [1], [1, 2]]

條件：無

任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。

步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S091





S093 · implies操作前

next ORIGINAL basis index=3;basis set=[2];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [1, 2], [0, 1, 2]]

操作後

ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [2], [3]], [[0], [0, 2], [0, 3]], [[1], [1, 2], [1, 3]], [[0, 1], [0, 1, 2], [0, 1, 3]], [[1, 2], [1, 2], [2, 3]], [[0, 1, 2], [0, 1, 2], [0, 2, 3]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]]

條件：無

任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。

步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S092





S094 · implies操作前

next ORIGINAL basis index=4;basis set=[3];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]

操作後

ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [3], [4]], [[0], [0, 3], [0, 4]], [[1], [1, 3], [1, 4]], [[0, 1], [0, 1, 3], [0, 1, 4]], [[2], [2, 3], [3, 4]], [[0, 2], [0, 2, 3], [0, 3, 4]], [[1, 2], [1, 2, 3], [2, 4]], [[0, 1, 2], [0, 1, 2, 3], [0, 2, 4]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 3], [0, 2], [0, 2, 3], [0, 3], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]]

條件：無

任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。

步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S093





S095 · implies操作前

next ORIGINAL basis index=5;basis set=[3, 4];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]

操作後

ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [3, 4], [5]], [[0], [0, 3, 4], [0, 5]], [[1], [1, 3, 4], [1, 5]], [[0, 1], [0, 1, 3, 4], [0, 1, 5]], [[2], [2, 3, 4], [3, 5]], [[0, 2], [0, 2, 3, 4], [0, 3, 5]], [[1, 2], [1, 2, 3, 4], [2, 5]], [[0, 1, 2], [0, 1, 2, 3, 4], [0, 2, 5]], [[3], [3, 4], [4, 5]], [[0, 3], [0, 3, 4], [0, 4, 5]], [[1, 3], [1, 3, 4], [1, 4, 5]], [[0, 1, 3], [0, 1, 3, 4], [0, 1, 4, 5]], [[2, 3], [2, 3, 4], [3, 4, 5]], [[0, 2, 3], [0, 2, 3, 4], [0, 3, 4, 5]], [[1, 2, 3], [1, 2, 3, 4], [2, 4, 5]], [[0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 4, 5]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 3], [0, 3, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 3], [1, 3, 4], [2], [2, 3], [2, 3, 4], [3], [3, 4]]

條件：無

任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。

步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S094





S096 · implies操作前

next ORIGINAL basis index=6;basis set=[4];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3], [3, 4], [0, 3, 4], [1, 3, 4], [0, 1, 3, 4], [2, 3, 4], [0, 2, 3, 4], [1, 2, 3, 4], [0, 1, 2, 3, 4]]

操作後

ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [4], [6]], [[0], [0, 4], [0, 6]], [[1], [1, 4], [1, 6]], [[0, 1], [0, 1, 4], [0, 1, 6]], [[2], [2, 4], [3, 6]], [[0, 2], [0, 2, 4], [0, 3, 6]], [[1, 2], [1, 2, 4], [2, 6]], [[0, 1, 2], [0, 1, 2, 4], [0, 2, 6]], [[3], [3, 4], [4, 6]], [[0, 3], [0, 3, 4], [0, 4, 6]], [[1, 3], [1, 3, 4], [1, 4, 6]], [[0, 1, 3], [0, 1, 3, 4], [0, 1, 4, 6]], [[2, 3], [2, 3, 4], [3, 4, 6]], [[0, 2, 3], [0, 2, 3, 4], [0, 3, 4, 6]], [[1, 2, 3], [1, 2, 3, 4], [2, 4, 6]], [[0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 4, 6]], [[3, 4], [3, 4], [5, 6]], [[0, 3, 4], [0, 3, 4], [0, 5, 6]], [[1, 3, 4], [1, 3, 4], [1, 5, 6]], [[0, 1, 3, 4], [0, 1, 3, 4], [0, 1, 5, 6]], [[2, 3, 4], [2, 3, 4], [3, 5, 6]], [[0, 2, 3, 4], [0, 2, 3, 4], [0, 3, 5, 6]], [[1, 2, 3, 4], [1, 2, 3, 4], [2, 5, 6]], [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], [0, 2, 5, 6]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]

條件：無

任意新prefix子族不是不選新basis元素，就是選它並與前prefix的某union合併，因此兩類選擇完整且每一狀態有真index見證。只合併相同union，不去掉原basis位置；狀態最多2^n，不需假裝抽樣了2^basisCount個族。

步驟識別與依賴規則：topbasis_complete_prefix_union_DP_not_sampled_subfamilies · 依賴：G001, S095





S097 · implies操作前

ALL original basis positions processed;distinct union states=32<=2^5;canonical ALL-union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]

操作後

prefix induction covers EVERY original-index subfamily and every state is its actual union;mathematical all-union family COMPLETE including empty union

條件：無

以每步的omit/include歸納證所有子族的union都在狀態中，反向每個狀態有實際子族見證，所以雙向完備。沒有把較少狀態數說成逐次枚舉的subfamily次數。

步驟識別與依賴規則：topbasis_all_subfamily_unions_complete_not_exponential_enumeration_claim · 依賴：G001, S096





S098 · implies操作前

JUST-VERIFIED cover and ALL local refinement for ORIGINAL basis=[[0], [1], [1, 2], [2], [3], [3, 4], [4]];COMPLETE all-union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]

操作後

empty/full carrier present;ANY union of basis-unions flattens to a basis-union;for x in U intersection V choose basis B,C around x then its verified local refiner inside B intersection C,so U intersection V is itself a basis-union;this family IS a topology on Fin(5)

條件：無

局部細化讓每個交集點都有落在U∩V中的basis元素，全部這些basis的union恰為U∩V；這才證交閉包，不要求basis族本身交封閉。加上任意union flatten、cover給全集及空union給空集，得到完整topology公理。

步驟識別與依賴規則：topbasis_basis_to_topology_arbitrary_union_and_local_intersection_proof · 依賴：G001, S097





S099 · implies操作前

ORIGINAL basis membership rows[basis_index,given_open_index_or_null]=[[0, 1], [1, 17], [2, 18], [3, 25], [4, 29], [5, 30], [6, 31]];given candidate-open generation rows[open_index,certified_union_index_or_null]=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 6], [7, 7], [8, 8], [9, 9], [10, 10], [11, 11], [12, 12], [13, 13], [14, 14], [15, 15], [16, 16], [17, 17], [18, 18], [19, 19], [20, 20], [21, 21], [22, 22], [23, 23], [24, 24], [25, 25], [26, 26], [27, 27], [28, 28], [29, 29], [30, 30], [31, 31]]

操作後

basis_elements_open=是;every_given_open_is_basis_union=是;source component agrees with actual mathematical union predicate=是

條件：無

basis元素列為給定open與給定每個open由basis生成是反方向的兩項責任；相對某個topology的basis要兩者都過，不能只看抽象basis資格。

步驟識別與依賴規則：topbasis_given_openness_and_every_open_generation_separate · 依賴：G001, S098





S100 · implies操作前

ACTUAL topology_valid=是,abstract_basis_valid=是,basis_elements_open=是,every_open_generated=是

操作後

basis_for_THIS_given_topology=是 is the FOUR-WAY conjunction;any passing component alone is insufficient

條件：無

四項合取才完成這個既有topology的basis審查，保原candidate族與原旗標；沒有升格成任意topology或未給空間的結論。

步驟識別與依賴規則：topbasis_basis_for_given_topology_all_responsibilities_conjunction · 依賴：G001, S099









## 逐步反向核對


- S100 · 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。
- S099 · 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。
- S098 · 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。
- S097 · 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。
- S096 · 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。
- S095 · 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。
- S094 · 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。
- S093 · 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。
- S092 · 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。
- S091 · 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。
- S090 · 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。
- S089 · 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。
- S088 · 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。
- S087 · 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。
- S086 · 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。
- S085 · 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。
- S084 · 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。
- S083 · 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。
- S082 · 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。
- S081 · 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。
- S080 · 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。
- S079 · 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。
- S078 · 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。
- S077 · 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。
- S076 · 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。
- S075 · 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。
- S074 · 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。
- S073 · 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。
- S072 · 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。
- S071 · 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。
- S070 · 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。
- S069 · 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。
- S068 · 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。
- S067 · 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。
- S066 · 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。
- S065 · 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。
- S064 · 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。
- S063 · 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。
- S062 · 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。
- S061 · 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。
- S060 · 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。
- S059 · 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。
- S058 · 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。
- S057 · 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。
- S056 · 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。
- S055 · 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。
- S054 · 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。
- S053 · 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。
- S052 · 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。
- S051 · 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。
- S050 · 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。
- S049 · 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。
- S048 · 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。
- S047 · 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。
- S046 · 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。
- S045 · 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。
- S044 · 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。
- S043 · 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。
- S042 · 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。
- S041 · 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。
- S040 · 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。
- S039 · 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。
- S038 · 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。
- S037 · 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。
- S036 · 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。
- S035 · 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。
- S034 · 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。
- S033 · 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。
- S032 · 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。
- S031 · 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。
- S030 · 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。
- S029 · 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。
- S028 · 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。
- S027 · 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。
- S026 · 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。
- S025 · 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。
- S024 · 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。
- S023 · 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。
- S022 · 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。
- S021 · 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。
- S020 · 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。
- S019 · 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。
- 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
- 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

(是, 是, 是, 是, 是)

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: 05ec14679e25a3843267b11cc55ccf1ea90c129fbd8f4e7dd0562f7d4906a9ff
- candidate record SHA-256: 5057053da351374dee50c4fa17a05e18f0ddd0aa22b70900be23b4da38be483d
- semantic review receipt SHA-256: c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d
- question-first witness SHA-256: e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7
- Lean: NOT_RUN · human review: NOT_PERFORMED

[同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261107-r03-topological-basis-0071.json)