Question 9275: Compare Two Qualified Topologies on the Same Finite Set

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0017-09275 · q-20261013-advance-20261018-r01-topological-space-0100

Original question and givens

Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], B=[[], [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]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.

G001 — Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], B=[[], [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]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes.

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-space

Step-by-step derivation

S001 · implies

Before

space=A;n=4;raw family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

universe=[0, 1, 2, 3];canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

Conditions: none

Type every candidate as a subset of the same Fin(n),deduplicate elements and members,and order by binary masks.The open family and underlying set are distinct data layers.

Step identifiers and dependencies

Rule: finite_topology_normalize_relative_universe · Uses: G001

S002 · implies

Before

space=A;empty=Yes;full=Yes;union closed=Yes;intersection closed=Yes

After

topology=Yes;first union failure=none;first intersection failure=none

Conditions: none

All four conditions are required.On a finite underlying set there are finitely many distinct opens,so an arbitrary union reduces to a finite distinct-member union,covered by binary closure and the empty union.The empty intersection needs the whole set.Vacuous binary closure does not waive empty/whole-set membership.

Step identifiers and dependencies

Rule: finite_topology_axioms_joint_gate · Uses: G001, S001

S003 · implies

Before

space=B;n=4;raw family=[[], [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

universe=[0, 1, 2, 3];canonical opens=[[], [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]]

Conditions: none

Type every candidate as a subset of the same Fin(n),deduplicate elements and members,and order by binary masks.The open family and underlying set are distinct data layers.

Step identifiers and dependencies

Rule: finite_topology_normalize_relative_universe · Uses: G001, S002

S004 · implies

Before

space=B;empty=Yes;full=Yes;union closed=Yes;intersection closed=Yes

After

topology=Yes;first union failure=none;first intersection failure=none

Conditions: none

Step identifiers and dependencies

Rule: finite_topology_axioms_joint_gate · Uses: G001, S003

S005 · implies

Before

direction=A in B;open member=[];other family=[[], [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

member included=Yes

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S004

S006 · implies

Before

direction=A in B;open member=[0];other family=[[], [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

member included=Yes

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S005

S007 · implies

Before

direction=A in B;open member=[0, 1];other family=[[], [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

member included=Yes

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S006

S008 · implies

Before

direction=A in B;open member=[0, 1, 2];other family=[[], [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

member included=Yes

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S007

S009 · implies

Before

direction=A in B;open member=[0, 1, 2, 3];other family=[[], [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

member included=Yes

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S008

S010 · implies

Before

direction=B in A;open member=[];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=Yes

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S009

S011 · implies

Before

direction=B in A;open member=[0];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=Yes

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S010

S012 · implies

Before

direction=B in A;open member=[1];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=No

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S011

S013 · implies

Before

direction=B in A;open member=[0, 1];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=Yes

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S012

S014 · implies

Before

direction=B in A;open member=[2];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=No

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S013

S015 · implies

Before

direction=B in A;open member=[0, 2];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=No

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S014

S016 · implies

Before

direction=B in A;open member=[1, 2];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=No

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S015

S017 · implies

Before

direction=B in A;open member=[0, 1, 2];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=Yes

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S016

S018 · implies

Before

direction=B in A;open member=[3];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=No

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S017

S019 · implies

Before

direction=B in A;open member=[0, 3];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=No

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S018

S020 · implies

Before

direction=B in A;open member=[1, 3];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=No

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S019

S021 · implies

Before

direction=B in A;open member=[0, 1, 3];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=No

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S020

S022 · implies

Before

direction=B in A;open member=[2, 3];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=No

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S021

S023 · implies

Before

direction=B in A;open member=[0, 2, 3];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=No

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S022

S024 · implies

Before

direction=B in A;open member=[1, 2, 3];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=No

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S023

S025 · implies

Before

direction=B in A;open member=[0, 1, 2, 3];other family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]

After

member included=Yes

Conditions: none

Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here.

Step identifiers and dependencies

Rule: topology_open_family_inclusion_member · Uses: G001, S024

S026 · implies

Before

A subset B=Yes;B subset A=No

After

topology relation=A coarser

Conditions: none

If every A-open is B-open,B is finer and A coarser.Both inclusions mean equality;neither means incomparable.

Step identifiers and dependencies

Rule: finer_coarser_direction · Uses: G001, S025

Reverse checks

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

(4, [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [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]], Yes, No, A-coarser)

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: bb9c2856742b4d4aec5a370529d4f1dae2f85ed05fecbb3a6f3a17472a9bd9d4
candidate record SHA-256: f7b096e4a47bbe1fc4a0946a0de2004c867716d0c84eabf934f21d98aff2152e
semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
Lean: NOT_RUN · human review: NOT_PERFORMED

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