# Question 9242: Topology Qualification before Discrete or Indiscrete Classification

[Return to catalogue page 47](https://ocme.evemisslab.com/en/questions/page/47/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261018-r01-topological-space-foundations/#q-20261013-advance-20261018-r01-topological-space-0067)

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.

R0017-09242 · q-20261013-advance-20261018-r01-topological-space-0067



## Original question and givens

Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [0, 2], [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, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.


- G001 — Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [2], [0, 2], [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, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.





## 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=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]

After

universe=[0, 1, 2, 3];canonical opens=[[], [2], [0, 2], [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;canonical family=[[], [2], [0, 2], [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 dependenciesRule: binary_set_operation_symmetry_coverage · Uses: G001, S001





S003 · impliesBefore

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 dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S002





S004 · impliesBefore

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

After

union=[0, 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 dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S003





S005 · impliesBefore

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

After

union=[0, 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 dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S004





S006 · impliesBefore

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 dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S005





S007 · impliesBefore

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

After

union=[0, 2];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 dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S006





S008 · impliesBefore

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

After

union=[0, 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 dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S007





S009 · impliesBefore

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 dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S008





S010 · impliesBefore

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

After

union=[0, 2, 3];in family=Yes;intersection=[0, 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 dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S009





S011 · impliesBefore

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

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0, 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 dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S010





S012 · impliesBefore

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 dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S011





S013 · 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, S012





S014 · impliesBefore

topology qualified=Yes;family size=5;powerset size=16

After

classification=other topology

Conditions: none

Qualify the topology first.All2^n subsets give the discrete topology;otherwise only empty and whole give the indiscrete topology.On the empty space both coincide,and the protocol classifies discrete first.

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









## Reverse checks


- 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, [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], Yes, other-topology)

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: 0578fc996059c17c2b4271341d5ed27c3006b78807ec88fdd909b19f1676bf0f
- candidate record SHA-256: d9a25a5633b1370f3b8b2d26113b5623be93c360473999cb799ede3150aecc87
- 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-0067.json)