# Question 9296: Every Open, Closed, and Clopen Set in a Finite 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.

R0017-09296 · q-20261013-advance-20261018-r01-topological-space-0046



## 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. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.


- 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. This family is required to be an actual topology. Give (n, open family, closed family, clopen family), using complements relative to X.





## 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](https://ocme.evemisslab.com/en/mko/mko-topological-space/)





## Step-by-step derivation

S001 · impliesBefore

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 dependenciesRule: finite_topology_normalize_relative_universe · Uses: G001





S002 · impliesBefore

space=given;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 dependenciesRule: finite_topology_axioms_joint_gate · Uses: G001, S001





S003 · impliesBefore

universe Fin(4);open=[]

After

relative complement=[0, 1, 2, 3];closed=Yes

Conditions: none

A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open.

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





S004 · impliesBefore

universe Fin(4);open=[0]

After

relative complement=[1, 2, 3];closed=Yes

Conditions: none

A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open.

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





S005 · impliesBefore

universe Fin(4);open=[0, 3]

After

relative complement=[1, 2];closed=Yes

Conditions: none

A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open.

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





S006 · impliesBefore

universe Fin(4);open=[0, 2, 3]

After

relative complement=[1];closed=Yes

Conditions: none

A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open.

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





S007 · impliesBefore

universe Fin(4);open=[0, 1, 2, 3]

After

relative complement=[];closed=Yes

Conditions: none

A closed set is the complement of an open set in this same underlying set;it is not defined as a set that is not open.

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





S008 · impliesBefore

opens=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]];closed sets=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]

After

clopen sets=[[], [0, 1, 2, 3]]

Conditions: none

Exactly the members of both families are clopen.Empty and whole sets are included,and other sets can also be clopen.

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









## Reverse checks


- 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]], [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

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: 6b66a6f74bf9856f1953cf028d7d7567c3d0e9b62d8666edd1c63e62abb7d599
- candidate record SHA-256: 042fe4d82463d9c408180d0a91af65585343f5693063d402ca9b0c6f6d8540b7
- semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
- question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261018-r01-topological-space-0046.json)