# Question 20647: A Basis for Some Topology Need Not Generate This Given Topology

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0040-20647 · q-20261014-advance-20261107-r03-topological-basis-0072



## Original question and givens

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete 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]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).


- G001 — Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete 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]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-topological-basis](https://ocme.evemisslab.com/en/mko/mko-topological-basis/)





## Step-by-step derivation

S001 · impliesBefore

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

After

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

Conditions: none

Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

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

After

within-set numeric normalization=[[0], [0, 1], [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

Conditions: none

A topological basis is a family of subsets,not a vector-space basis;its size may exceed n.Normalize point order within each set without changing original family indices or deleting duplicates to repair data.An empty candidate family can still be audited.

Step identifiers and dependenciesRule: topbasis_canonical_candidate_family_not_semantic_reordering · Uses: G001, S001





S003 · impliesBefore

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

After

ALL [point,containing_basis_indices] rows=[[0, [0, 1]], [1, [1, 2, 3]], [2, [3, 4]], [3, [5, 6]], [4, [6, 7]]];covers WHOLE carrier=Yes

Conditions: none

Every carrier point needs a containing candidate element for coverage.This does not replace local intersection refinement.With n≥1 an empty family fails coverage rather than being discarded as an unauditable candidate.

Step identifiers and dependenciesRule: topbasis_every_point_cover_separate_from_refinement · Uses: G001, S002





S004 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S003





S005 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S004





S006 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S005





S007 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[0, 3];actual basis elements=[[0], [1, 2]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S006





S008 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[0, 4];actual basis elements=[[0], [2]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S007





S009 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S008





S010 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[0, 6];actual basis elements=[[0], [3, 4]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S009





S011 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[0, 7];actual basis elements=[[0], [4]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S010





S012 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S011





S013 · impliesBefore

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

After

ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0, 1]], [1, [1, 2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S012





S014 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S013





S015 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[1, 3];actual basis elements=[[0, 1], [1, 2]];FULL intersection=[1]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S014





S016 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[1, 4];actual basis elements=[[0, 1], [2]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S015





S017 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[1, 5];actual basis elements=[[0, 1], [3]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S016





S018 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[1, 6];actual basis elements=[[0, 1], [3, 4]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S017





S019 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[1, 7];actual basis elements=[[0, 1], [4]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S018





S020 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S019





S021 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S020





S022 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S021





S023 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S022





S024 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S023





S025 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S024





S026 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[2, 6];actual basis elements=[[1], [3, 4]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S025





S027 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[2, 7];actual basis elements=[[1], [4]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S026





S028 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[3, 0];actual basis elements=[[1, 2], [0]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S027





S029 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[3, 1];actual basis elements=[[1, 2], [0, 1]];FULL intersection=[1]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S028





S030 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S029





S031 · impliesBefore

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

After

ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2, 3]], [2, [3, 4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S030





S032 · impliesBefore

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

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S031





S033 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S032





S034 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[3, 6];actual basis elements=[[1, 2], [3, 4]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S033





S035 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[3, 7];actual basis elements=[[1, 2], [4]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S034





S036 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[4, 0];actual basis elements=[[2], [0]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S035





S037 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[4, 1];actual basis elements=[[2], [0, 1]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S036





S038 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S037





S039 · impliesBefore

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

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S038





S040 · impliesBefore

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

After

ALL [intersection_point,ALL refining original basis indices] rows=[[2, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S039





S041 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S040





S042 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S041





S043 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[4, 7];actual basis elements=[[2], [4]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S042





S044 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S043





S045 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[5, 1];actual basis elements=[[3], [0, 1]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S044





S046 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S045





S047 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S046





S048 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S047





S049 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S048





S050 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S049





S051 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[5, 7];actual basis elements=[[3], [4]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S050





S052 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[6, 0];actual basis elements=[[3, 4], [0]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S051





S053 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[6, 1];actual basis elements=[[3, 4], [0, 1]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S052





S054 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[6, 2];actual basis elements=[[3, 4], [1]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S053





S055 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[6, 3];actual basis elements=[[3, 4], [1, 2]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S054





S056 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S055





S057 · impliesBefore

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

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S056





S058 · impliesBefore

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

After

ALL [intersection_point,ALL refining original basis indices] rows=[[3, [5, 6]], [4, [6, 7]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S057





S059 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[6, 7];actual basis elements=[[3, 4], [4]];FULL intersection=[4]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[4, [7]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S058





S060 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[7, 0];actual basis elements=[[4], [0]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S059





S061 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[7, 1];actual basis elements=[[4], [0, 1]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S060





S062 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[7, 2];actual basis elements=[[4], [1]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S061





S063 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[7, 3];actual basis elements=[[4], [1, 2]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S062





S064 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[7, 4];actual basis elements=[[4], [2]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S063





S065 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[7, 5];actual basis elements=[[4], [3]];FULL intersection=[]

After

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

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S064





S066 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[7, 6];actual basis elements=[[4], [3, 4]];FULL intersection=[4]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[4, [7]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S065





S067 · impliesBefore

ORIGINAL ordered basis positions(left,right)=[7, 7];actual basis elements=[[4], [4]];FULL intersection=[4]

After

ALL [intersection_point,ALL refining original basis indices] rows=[[4, [7]]];empty intersection gives VACUOUS local obligation;first failure retained in source order

Conditions: none

For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.

Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S066





S068 · impliesBefore

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

After

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

Conditions: none

Their conjunction gives an abstract basis.Coverage alone is insufficient,and a valid basis need not be intersection-closed.A basis for a separately supplied topology also needs openness and generation of every given open.

Step identifiers and dependenciesRule: topbasis_cover_and_local_refinement_conjunction · Uses: G001, S067





S069 · impliesBefore

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)

After

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

Conditions: none

A topological basis is a family of subsets,not a vector-space basis;its size may exceed n.Normalize point order within each set without changing original family indices or deleting duplicates to repair data.An empty candidate family can still be audited.

Step identifiers and dependenciesRule: topbasis_canonical_candidate_family_not_semantic_reordering · Uses: G001, S068





S070 · impliesBefore

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)

After

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

Conditions: none

This operation audits a given candidate family,retaining false flags for invalid candidates.Structural qualification alone does not establish a topology,and blanket rejection would omit the requested audit.

Step identifiers and dependenciesRule: topbasis_given_topology_candidate_not_blanket_assumed_valid · Uses: G001, S069





S071 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S070





S072 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S071





S073 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S072





S074 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S073





S075 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S074





S076 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S075





S077 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S076





S078 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S077





S079 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S078





S080 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S079





S081 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S080





S082 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S081





S083 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S082





S084 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S083





S085 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S084





S086 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S085





S087 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S086





S088 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S087





S089 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S088





S090 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S089





S091 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S090





S092 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S091





S093 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S092





S094 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S093





S095 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S094





S096 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S095





S097 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S096





S098 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S097





S099 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S098





S100 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S099





S101 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S100





S102 · impliesBefore

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

After

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

Conditions: none

Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.

Step identifiers and dependenciesRule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S101





S103 · impliesBefore

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

After

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

Conditions: none

A finite family has only finitely many distinct opens,so arbitrary indexed unions reduce to finite unions.Apply that topology certificate only when empty/full sets and all closure obligations pass.For false candidates retain the actual failure without changing data.

Step identifiers and dependenciesRule: topbasis_given_topology_full_axiom_conjunction_not_only_binary_slogan · Uses: G001, S102





S104 · impliesBefore

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

After

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

Conditions: none

The empty subfamily always has empty union,without requiring the empty set to be a basis element.This step constructs unions only and does not call an invalid basis's unions a topology.

Step identifiers and dependenciesRule: topbasis_empty_union_initial_state_not_missing_empty_set · Uses: G001, S103





S105 · impliesBefore

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

After

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]]

Conditions: none

Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.

Step identifiers and dependenciesRule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S104





S106 · impliesBefore

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

After

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

Conditions: none

Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.

Step identifiers and dependenciesRule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S105





S107 · impliesBefore

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

After

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

Conditions: none

Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.

Step identifiers and dependenciesRule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S106





S108 · impliesBefore

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

After

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

Conditions: none

Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.

Step identifiers and dependenciesRule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S107





S109 · impliesBefore

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

After

ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [2], [4]], [[0], [0, 2], [0, 4]], [[1], [1, 2], [2, 4]], [[0, 1], [0, 1, 2], [1, 4]], [[1, 2], [1, 2], [3, 4]], [[0, 1, 2], [0, 1, 2], [0, 3, 4]]];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]]

Conditions: none

Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.

Step identifiers and dependenciesRule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S108





S110 · impliesBefore

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

After

ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [3], [5]], [[0], [0, 3], [0, 5]], [[1], [1, 3], [2, 5]], [[0, 1], [0, 1, 3], [1, 5]], [[2], [2, 3], [4, 5]], [[0, 2], [0, 2, 3], [0, 4, 5]], [[1, 2], [1, 2, 3], [3, 5]], [[0, 1, 2], [0, 1, 2, 3], [0, 3, 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, 3], [0, 2], [0, 2, 3], [0, 3], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]]

Conditions: none

Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.

Step identifiers and dependenciesRule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S109





S111 · impliesBefore

next ORIGINAL basis index=6;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]]

After

ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [3, 4], [6]], [[0], [0, 3, 4], [0, 6]], [[1], [1, 3, 4], [2, 6]], [[0, 1], [0, 1, 3, 4], [1, 6]], [[2], [2, 3, 4], [4, 6]], [[0, 2], [0, 2, 3, 4], [0, 4, 6]], [[1, 2], [1, 2, 3, 4], [3, 6]], [[0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 6]], [[3], [3, 4], [5, 6]], [[0, 3], [0, 3, 4], [0, 5, 6]], [[1, 3], [1, 3, 4], [2, 5, 6]], [[0, 1, 3], [0, 1, 3, 4], [1, 5, 6]], [[2, 3], [2, 3, 4], [4, 5, 6]], [[0, 2, 3], [0, 2, 3, 4], [0, 4, 5, 6]], [[1, 2, 3], [1, 2, 3, 4], [3, 5, 6]], [[0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 3, 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, 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]]

Conditions: none

Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.

Step identifiers and dependenciesRule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S110





S112 · impliesBefore

next ORIGINAL basis index=7;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]]

After

ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [4], [7]], [[0], [0, 4], [0, 7]], [[1], [1, 4], [2, 7]], [[0, 1], [0, 1, 4], [1, 7]], [[2], [2, 4], [4, 7]], [[0, 2], [0, 2, 4], [0, 4, 7]], [[1, 2], [1, 2, 4], [3, 7]], [[0, 1, 2], [0, 1, 2, 4], [0, 3, 7]], [[3], [3, 4], [5, 7]], [[0, 3], [0, 3, 4], [0, 5, 7]], [[1, 3], [1, 3, 4], [2, 5, 7]], [[0, 1, 3], [0, 1, 3, 4], [1, 5, 7]], [[2, 3], [2, 3, 4], [4, 5, 7]], [[0, 2, 3], [0, 2, 3, 4], [0, 4, 5, 7]], [[1, 2, 3], [1, 2, 3, 4], [3, 5, 7]], [[0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 3, 5, 7]], [[3, 4], [3, 4], [6, 7]], [[0, 3, 4], [0, 3, 4], [0, 6, 7]], [[1, 3, 4], [1, 3, 4], [2, 6, 7]], [[0, 1, 3, 4], [0, 1, 3, 4], [1, 6, 7]], [[2, 3, 4], [2, 3, 4], [4, 6, 7]], [[0, 2, 3, 4], [0, 2, 3, 4], [0, 4, 6, 7]], [[1, 2, 3, 4], [1, 2, 3, 4], [3, 6, 7]], [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], [0, 3, 6, 7]]];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]]

Conditions: none

Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.

Step identifiers and dependenciesRule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S111





S113 · impliesBefore

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]]

After

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

Conditions: none

Omit/include induction puts every subfamily union into the states,and every state has an actual subfamily witness,proving both directions.Do not misreport the smaller distinct-state count as the number of subfamilies explicitly enumerated.

Step identifiers and dependenciesRule: topbasis_all_subfamily_unions_complete_not_exponential_enumeration_claim · Uses: G001, S112





S114 · impliesBefore

JUST-VERIFIED cover and ALL local refinement for ORIGINAL basis=[[0], [0, 1], [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]]

After

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)

Conditions: none

Local refinement supplies a basis element around each point inside U∩V,and their union equals that intersection.This proves intersection closure without requiring the basis family itself intersection-closed.Arbitrary-union flattening,coverage of the full carrier and the empty union complete the topology axioms.

Step identifiers and dependenciesRule: topbasis_basis_to_topology_arbitrary_union_and_local_intersection_proof · Uses: G001, S113





S115 · impliesBefore

ORIGINAL basis membership rows[basis_index,given_open_index_or_null]=[[0, 1], [1, 2], [2, 17], [3, 18], [4, 25], [5, 29], [6, 30], [7, 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]]

After

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

Conditions: none

Basis elements being listed as given opens and every given open being generated by the basis are opposite-direction obligations.A basis for this supplied topology needs both,not just abstract basis validity.

Step identifiers and dependenciesRule: topbasis_given_openness_and_every_open_generation_separate · Uses: G001, S114





S116 · impliesBefore

ACTUAL topology_valid=Yes,abstract_basis_valid=Yes,basis_elements_open=Yes,every_open_generated=Yes

After

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

Conditions: none

Only their conjunction establishes a basis for this given topology.Retain the original candidate families and flags without promoting a passing component to an arbitrary-space conclusion.

Step identifiers and dependenciesRule: topbasis_basis_for_given_topology_all_responsibilities_conjunction · Uses: G001, S115









## Reverse checks


- S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(Yes, Yes, Yes, Yes, Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 921402531362790cf607c1b386799bca0293ec4a783f8ffef0dd1d2639d7378d
- candidate record SHA-256: 3d61cae4bc0bafa19a2b0e7a8074fe5614f0ab332440de0549c653979060f53b
- semantic review receipt SHA-256: c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d
- question-first witness SHA-256: e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261107-r03-topological-basis-0072.json)