# Question 9846: Relative Complement and Open/Closed Duality

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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.

R0018-09846 · q-20261013-advance-20261019-r01-open-closed-set-0046



## Original question and givens

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).


- G001 — Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).





## 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-open-closed-set](https://ocme.evemisslab.com/en/mko/mko-open-closed-set/)





## Step-by-step derivation

S001 · impliesBefore

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

After

universe=[0, 1, 2, 3];normalized opens=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]];empty=Yes;full=Yes;union closed=Yes;intersection closed=Yes

Conditions: none

Deduplicate subsets/families and order by masks.Check empty/whole membership and all binary union/intersection closure.On a finite underlying set,arbitrary unions reduce to finitely many distinct members.These tasks require an actual topology;invalid data is not used for later properties.

Step identifiers and dependenciesRule: given_complete_finite_topology_guard · Uses: G001





S002 · impliesBefore

universe=[0, 1, 2, 3];candidate=[2]

After

relative complement=[0, 1, 3];candidate open=No;complement open=No

Conditions: none

After typing the candidate,take its complement in the fixed universe.Check openness of the candidate and complement independently;not-open is not closed.

Step identifiers and dependenciesRule: relative_complement_open_test · Uses: G001, S001





S003 · impliesBefore

complement open=No;candidate closed=No

After

duality equality=Yes

Conditions: none

Closed is defined by openness of the same relative complement;this equality does not identify closed with negated open.

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









## Reverse checks


- 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, 1, 3], No, No, No, Yes)

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: eda22c666e936d93798baed3b7e4e0af7a67cd777b5e6760b49cc3654e3f8b9e
- candidate record SHA-256: ec4433a8344d817cfbc71ca60393911af33997841d372dc205738e38f09be463
- semantic review receipt SHA-256: bd35da62087a15d6b89b45585dc46e89b9efbe097cba7b3c707b5030496f013c
- question-first witness SHA-256: fab45a10d8ac95c7d61d32e47d1bdd3ca862c17c3289a6bf4b01d4dc9492fbb6
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261019-r01-open-closed-set-0046.json)