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Question 9221: Complete Finite Topology Axiom Audit

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0017-09221 · q-20261013-advance-20261018-r01-topological-space-0021

Original question and givens

Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[0], [0, 3], [0, 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], [0, 3], [0, 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.

Step-by-step derivation

S001 · implies

Before

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

After

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

After

union=[0, 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, S002

S004 · implies

Before

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

After

union=[0, 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, S003

S005 · 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, S004

S006 · implies

Before

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

After

union=[0, 2, 3];in family=Yes;intersection=[0, 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, S005

S007 · implies

Before

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

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0, 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, S006

S008 · implies

Before

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

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0, 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, S007

S009 · implies

Before

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

After

topology=No;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, S008

Reverse checks

  • 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, 3], [0, 2, 3], [0, 1, 2, 3]], No, Yes, Yes, Yes, No, none, 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: 7d1e09c970961ca9f1014df413732179ebb0cff14fdde6d4c8c275529ff8cdcc
  • candidate record SHA-256: 733fb7cd9345024cd12cb5c6bf06b875c3c62fabcb5e67db9152d01e1232f0f0
  • semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
  • question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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