Question 9223: Complete Finite Topology Axiom Audit

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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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R0017-09223 · q-20261013-advance-20261018-r01-topological-space-0023

Original question and givens

Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).

G001 — Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, contains empty set?, contains X?, all binary unions closed?, all binary intersections closed?, topology?, union counterexample or none, intersection counterexample or none).

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

After

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

Before

space=given;canonical family=[[], [0], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]

After

self pairs are idempotent;unordered distinct pairs cover all ordered pairs

Conditions: none

Self pairs are covered by A∪A=A and A∩A=A.Commutativity means all distinct unordered pairs cover every ordered pair;this is not sampling.

Step identifiers and dependencies

Rule: binary_set_operation_symmetry_coverage · Uses: G001, S001

S003 · implies

Before

space=given;A=[];B=[0]

After

union=[0];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S002

S004 · implies

Before

space=given;A=[];B=[2]

After

union=[2];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S003

S005 · implies

Before

space=given;A=[];B=[2, 3]

After

union=[2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S004

S006 · implies

Before

space=given;A=[];B=[1, 2, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S005

S007 · implies

Before

space=given;A=[];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S006

S008 · implies

Before

space=given;A=[0];B=[2]

After

union=[0, 2];in family=No;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S007

S009 · implies

Before

space=given;A=[0];B=[2, 3]

After

union=[0, 2, 3];in family=No;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S008

S010 · implies

Before

space=given;A=[0];B=[1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S009

S011 · implies

Before

space=given;A=[0];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S010

S012 · implies

Before

space=given;A=[2];B=[2, 3]

After

union=[2, 3];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S011

S013 · implies

Before

space=given;A=[2];B=[1, 2, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S012

S014 · implies

Before

space=given;A=[2];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S013

S015 · implies

Before

space=given;A=[2, 3];B=[1, 2, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[2, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S014

S016 · implies

Before

space=given;A=[2, 3];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[2, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S015

S017 · implies

Before

space=given;A=[1, 2, 3];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[1, 2, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S016

S018 · implies

Before

space=given;empty=Yes;full=Yes;union closed=No;intersection closed=Yes

After

topology=No;first union failure=[[0], [2], [0, 2]];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, S017

Reverse checks

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], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, Yes, No, Yes, No, [[0], [2], [0, 2]], none)

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: cdce05e788a228bc8c3abb4d2351e74b33e86a2f5418d591d40b330434cfa2f4
candidate record SHA-256: cffae26b83f0b69aa84939ddb5e84a0d98eca27015403e4c9c74216950827b5c
semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
Lean: NOT_RUN · human review: NOT_PERFORMED

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