Open, Closed, and Clopen Sets in Complete Finite Topologies
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Open and closed sets · qf-ext19-openclosed-family-closure · candidate
Canonical question JSONL · English JSONL · Validation report