# Open, Closed, and Clopen Sets in Complete Finite Topologies

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261013-advance-20261019-r01-open-closed-set-foundations · 2026-10-13 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 2], [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=[]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [], Yes, Yes, Yes, clopen)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [], Yes, Yes, Yes, clopen).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0001/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 2. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 2], [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=[0]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [0], No, No, No, neither)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0], No, No, No, neither).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0002/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 3. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 1, 2], [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=[1]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [1], No, No, No, neither)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1], No, No, No, neither).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0003/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 4. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 1, 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=[0, 1]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [0, 1], No, No, No, neither)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1], No, No, No, neither).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0004/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 5. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 1, 2], [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, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [2], No, No, No, neither)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [2], No, No, No, neither).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0005/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 6. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [0, 1, 2], [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=[0, 2]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [0, 2], No, No, No, neither)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 2], No, No, No, neither).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0006/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 7. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [0, 1, 2], [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=[1, 2]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [1, 2], Yes, No, No, open)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 2], Yes, No, No, open).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0007/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 8. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [0, 1, 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=[0, 1, 2]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [0, 1, 2], No, Yes, No, closed)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 2], No, Yes, No, closed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0008/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 9. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 1, 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=[3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [3], No, No, No, neither)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [3], No, No, No, neither).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0009/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 10. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [0, 1, 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=[0, 3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [0, 3], No, No, No, neither)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 3], No, No, No, neither).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0010/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 11. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 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=[1, 3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [1, 3], No, No, No, neither)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 3], No, No, No, neither).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0011/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 12. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 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=[0, 1, 3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [0, 1, 3], No, No, No, neither)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 3], No, No, No, neither).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0012/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 13. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 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, 3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [2, 3], No, Yes, No, closed)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [2, 3], No, Yes, No, closed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0013/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 14. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 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=[0, 2, 3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [0, 2, 3], No, Yes, No, closed)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 2, 3], No, Yes, No, closed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0014/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 15. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [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=[1, 2, 3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [1, 2, 3], No, No, No, neither)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 2, 3], No, No, No, neither).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0015/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 16. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [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=[0, 1, 2, 3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [0, 1, 2, 3], Yes, Yes, Yes, clopen)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 2, 3], Yes, Yes, Yes, clopen).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0016/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 17. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [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=[]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [], Yes, Yes, Yes, clopen)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [], Yes, Yes, Yes, clopen).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0017/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 18. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [1, 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=[0]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [0], No, Yes, No, closed)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0], No, Yes, No, closed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0018/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 19. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [1, 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=[1]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [1], No, No, No, neither)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1], No, No, No, neither).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0019/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 20. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [1, 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=[0, 1]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [0, 1], No, No, No, neither)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1], No, No, No, neither).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0020/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 21. Open, Closed, Both, or Neither in One Fixed Topology

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, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [2], No, No, No, neither)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [2], No, No, No, neither).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0021/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 22. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [1, 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=[0, 2]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [0, 2], No, Yes, No, closed)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 2], No, Yes, No, closed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0022/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 23. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [1, 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=[1, 2]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [1, 2], No, No, No, neither)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 2], No, No, No, neither).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0023/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 24. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [1, 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=[0, 1, 2]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(4, [0, 1, 2], No, Yes, No, closed)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 2], No, Yes, No, closed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0024/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 25. Open, Closed, Both, or Neither in One Fixed Topology

Use the complete finite topology on X=Fin(3)={0,…,2} with opens=[[], [0], [0, 1], [0, 1, 2]]. 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=[]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).

View answer and explanation(3, [], Yes, Yes, Yes, clopen)

Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [], Yes, Yes, Yes, clopen).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0025/)

Open and closed sets · qf-ext19-openclosed-status · candidate





## 26. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 2], [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=[]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [], [0, 1, 2, 3], Yes, Yes, Yes, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [], [0, 1, 2, 3], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0026/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 27. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 2], [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=[0]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [0], [1, 2, 3], No, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0], [1, 2, 3], No, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0027/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 28. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 1, 2], [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=[1]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [1], [0, 2, 3], No, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1], [0, 2, 3], No, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0028/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 29. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 1, 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=[0, 1]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [0, 1], [2, 3], No, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1], [2, 3], No, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0029/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 30. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 1, 2], [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?).

View answer and explanation(4, [2], [0, 1, 3], No, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [2], [0, 1, 3], No, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0030/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 31. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [0, 1, 2], [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=[0, 2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [0, 2], [1, 3], No, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 2], [1, 3], No, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0031/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 32. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [0, 1, 2], [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=[1, 2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [1, 2], [0, 3], Yes, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 2], [0, 3], Yes, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0032/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 33. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [0, 1, 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=[0, 1, 2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [0, 1, 2], [3], No, Yes, Yes, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 2], [3], No, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0033/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 34. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 1, 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=[3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [3], [0, 1, 2], No, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [3], [0, 1, 2], No, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0034/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 35. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [0, 1, 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=[0, 3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [0, 3], [1, 2], No, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 3], [1, 2], No, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0035/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 36. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 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=[1, 3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [1, 3], [0, 2], No, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 3], [0, 2], No, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0036/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 37. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 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=[0, 1, 3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [0, 1, 3], [2], No, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 3], [2], No, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0037/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 38. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 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, 3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [2, 3], [0, 1], No, Yes, Yes, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [2, 3], [0, 1], No, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0038/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 39. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 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=[0, 2, 3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [0, 2, 3], [1], No, Yes, Yes, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 2, 3], [1], No, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0039/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 40. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [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=[1, 2, 3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [1, 2, 3], [0], No, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 2, 3], [0], No, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0040/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 41. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [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=[0, 1, 2, 3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [0, 1, 2, 3], [], Yes, Yes, Yes, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 2, 3], [], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0041/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 42. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [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=[]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [], [0, 1, 2, 3], Yes, Yes, Yes, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [], [0, 1, 2, 3], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0042/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 43. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [1, 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=[0]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [0], [1, 2, 3], No, Yes, Yes, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0], [1, 2, 3], No, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0043/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 44. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [1, 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=[1]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [1], [0, 2, 3], No, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1], [0, 2, 3], No, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0044/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 45. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [1, 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=[0, 1]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [0, 1], [2, 3], No, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1], [2, 3], No, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0045/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 46. Relative Complement and Open/Closed Duality

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?).

View answer and explanation(4, [2], [0, 1, 3], No, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [2], [0, 1, 3], No, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0046/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 47. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [1, 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=[0, 2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [0, 2], [1, 3], No, Yes, Yes, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 2], [1, 3], No, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0047/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 48. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [1, 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=[1, 2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [1, 2], [0, 3], No, No, No, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 2], [0, 3], No, No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0048/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 49. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [1, 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=[0, 1, 2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(4, [0, 1, 2], [3], No, Yes, Yes, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 2], [3], No, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0049/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 50. Relative Complement and Open/Closed Duality

Use the complete finite topology on X=Fin(3)={0,…,2} with opens=[[], [0], [0, 1], [0, 1, 2]]. 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=[]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).

View answer and explanation(3, [], [0, 1, 2], Yes, Yes, Yes, Yes)

Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [], [0, 1, 2], Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0050/)

Open and closed sets · qf-ext19-openclosed-complement-duality · candidate





## 51. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 2], [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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0051/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 52. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 2], [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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0052/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 53. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 1, 2], [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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0053/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 54. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 1, 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0054/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 55. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 1, 2], [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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0055/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 56. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [0, 1, 2], [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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0056/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 57. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [0, 1, 2], [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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0057/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 58. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [0, 1, 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0058/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 59. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 1, 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0059/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 60. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [0, 1, 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0060/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 61. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0061/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 62. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0062/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 63. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0063/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 64. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0064/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 65. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0065/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 66. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0066/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 67. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0067/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 68. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [1, 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0068/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 69. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [1, 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0069/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 70. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [1, 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0070/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 71. Complete Open, Closed, and Clopen Families

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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(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]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (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]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0071/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 72. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [1, 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0072/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 73. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [1, 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0073/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 74. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [1, 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(4, [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0074/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 75. Complete Open, Closed, and Clopen Families

Use the complete finite topology on X=Fin(3)={0,…,2} with opens=[[], [0], [0, 1], [0, 1, 2]]. 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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

View answer and explanation(3, [[], [0], [0, 1], [0, 1, 2]], [[], [2], [1, 2], [0, 1, 2]], [[], [0, 1, 2]])

Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[], [0], [0, 1], [0, 1, 2]], [[], [2], [1, 2], [0, 1, 2]], [[], [0, 1, 2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0075/)

Open and closed sets · qf-ext19-openclosed-clopen-inventory · candidate





## 76. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 2], [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. mode=open_union, selected=[[], [0, 1], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, open_union, [[], [0, 1], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_union, [[], [0, 1], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0076/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 77. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 2], [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. mode=open_finite_intersection, selected=[[1], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, open_finite_intersection, [[1], [0, 1, 2]], Yes, [1], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_finite_intersection, [[1], [0, 1, 2]], Yes, [1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0077/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 78. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 1, 2], [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. mode=closed_intersection, selected=[[], [1, 3], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], Yes, [], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], Yes, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0078/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 79. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 1, 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. mode=closed_finite_union, selected=[[2], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, closed_finite_union, [[2], [0, 1, 2]], Yes, [0, 1, 2], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_finite_union, [[2], [0, 1, 2]], Yes, [0, 1, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0079/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 80. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 1, 2], [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. mode=open_union, selected=[[], [0, 2], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, open_union, [[], [0, 2], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_union, [[], [0, 2], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0080/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 81. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [0, 1, 2], [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. mode=open_finite_intersection, selected=[[1], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, open_finite_intersection, [[1], [0, 1, 2]], Yes, [1], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_finite_intersection, [[1], [0, 1, 2]], Yes, [1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0081/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 82. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [0, 1, 2], [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. mode=closed_intersection, selected=[[], [0, 3], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, closed_intersection, [[], [0, 3], [0, 1, 2, 3]], Yes, [], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_intersection, [[], [0, 3], [0, 1, 2, 3]], Yes, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0082/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 83. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [0, 1, 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. mode=closed_finite_union, selected=[[2], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, closed_finite_union, [[2], [0, 1, 2]], Yes, [0, 1, 2], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_finite_union, [[2], [0, 1, 2]], Yes, [0, 1, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0083/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 84. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 1, 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. mode=open_union, selected=[[], [0, 3], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, open_union, [[], [0, 3], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_union, [[], [0, 3], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0084/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 85. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [0, 1, 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. mode=open_finite_intersection, selected=[[1], [0, 1, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, open_finite_intersection, [[1], [0, 1, 3]], Yes, [1], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_finite_intersection, [[1], [0, 1, 3]], Yes, [1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0085/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 86. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 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. mode=closed_intersection, selected=[[], [0, 1], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, closed_intersection, [[], [0, 1], [0, 1, 2, 3]], Yes, [], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_intersection, [[], [0, 1], [0, 1, 2, 3]], Yes, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0086/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 87. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 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. mode=closed_finite_union, selected=[[1], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, closed_finite_union, [[1], [0, 1, 2]], Yes, [0, 1, 2], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_finite_union, [[1], [0, 1, 2]], Yes, [0, 1, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0087/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 88. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 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. mode=open_union, selected=[[], [0, 1], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, open_union, [[], [0, 1], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_union, [[], [0, 1], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0088/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 89. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 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. mode=open_finite_intersection, selected=[[1], [0, 1, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, open_finite_intersection, [[1], [0, 1, 3]], Yes, [1], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_finite_intersection, [[1], [0, 1, 3]], Yes, [1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0089/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 90. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [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. mode=closed_intersection, selected=[[], [1, 3], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], Yes, [], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], Yes, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0090/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 91. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [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. mode=closed_finite_union, selected=[[1], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, closed_finite_union, [[1], [0, 1, 2]], Yes, [0, 1, 2], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_finite_union, [[1], [0, 1, 2]], Yes, [0, 1, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0091/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 92. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [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. mode=open_union, selected=[[], [0, 2], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, open_union, [[], [0, 2], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_union, [[], [0, 2], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0092/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 93. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [1, 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. mode=open_finite_intersection, selected=[[1], [1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, open_finite_intersection, [[1], [1, 2, 3]], Yes, [1], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_finite_intersection, [[1], [1, 2, 3]], Yes, [1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0093/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 94. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [1, 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. mode=closed_intersection, selected=[[], [0, 3], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, closed_intersection, [[], [0, 3], [0, 1, 2, 3]], Yes, [], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_intersection, [[], [0, 3], [0, 1, 2, 3]], Yes, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0094/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 95. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [1, 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. mode=closed_finite_union, selected=[[0], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, closed_finite_union, [[0], [0, 1, 2]], Yes, [0, 1, 2], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_finite_union, [[0], [0, 1, 2]], Yes, [0, 1, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0095/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 96. Empty Families and Qualified Closure Premises

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. mode=open_union, selected=[[], [0, 3], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, open_union, [[], [0, 3], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_union, [[], [0, 3], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0096/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 97. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [1, 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. mode=open_finite_intersection, selected=[[1], [1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, open_finite_intersection, [[1], [1, 2, 3]], Yes, [1], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_finite_intersection, [[1], [1, 2, 3]], Yes, [1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0097/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 98. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [1, 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. mode=closed_intersection, selected=[[], [0, 1], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, closed_intersection, [[], [0, 1], [0, 1, 2, 3]], Yes, [], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_intersection, [[], [0, 1], [0, 1, 2, 3]], Yes, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0098/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 99. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [1, 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. mode=closed_finite_union, selected=[[0], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(4, closed_finite_union, [[0], [0, 1, 2]], Yes, [0, 1, 2], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_finite_union, [[0], [0, 1, 2]], Yes, [0, 1, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0099/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate





## 100. Empty Families and Qualified Closure Premises

Use the complete finite topology on X=Fin(3)={0,…,2} with opens=[[], [0], [0, 1], [0, 1, 2]]. 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. mode=open_union, selected=[[], [0, 1]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.

View answer and explanation(3, open_union, [[], [0, 1]], Yes, [0, 1], Yes)

Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, open_union, [[], [0, 1]], Yes, [0, 1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r01-open-closed-set-0100/)

Open and closed sets · qf-ext19-openclosed-family-closure · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261019-r01-open-closed-set-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261019-r01-open-closed-set-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261019-r01-open-closed-set-foundations/validation-report.json)