Point-Qualified Neighborhoods and All Open Witnesses
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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-r02-neighborhood-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. Point Guard and Every Open Neighborhood Witness
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. Point x=0, candidate N=[0, 1]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 0, Yes, [0, 1], [[0], [0, 1]], Yes)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1], [[0], [0, 1]], Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
2. Point Guard and Every Open Neighborhood Witness
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. Point x=1, candidate N=[2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 1, Yes, [2], [], No)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [2], [], No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
3. Point Guard and Every Open Neighborhood Witness
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. Point x=2, candidate N=[0, 2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 2, Yes, [0, 2], [[2], [0, 2]], Yes)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 2], [[2], [0, 2]], Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
4. Point Guard and Every Open Neighborhood Witness
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. Point x=3, candidate N=[1, 2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 3, Yes, [1, 2], [], No)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 2], [], No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
5. Point Guard and Every Open Neighborhood Witness
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. Point x=-1, candidate N=[0, 1, 2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, -1, No, [0, 1, 2], not-evaluated, not-evaluated)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0, 1, 2], not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
6. Point Guard and Every Open Neighborhood Witness
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. Point x=1, candidate N=[3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 1, Yes, [3], [], No)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [3], [], No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
7. Point Guard and Every Open Neighborhood Witness
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. Point x=2, candidate N=[0, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 2, Yes, [0, 3], [], No)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 3], [], No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
8. Point Guard and Every Open Neighborhood Witness
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. Point x=3, candidate N=[1, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 3, Yes, [1, 3], [[3], [1, 3]], Yes)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 3], [[3], [1, 3]], Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
9. Point Guard and Every Open Neighborhood Witness
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. Point x=0, candidate N=[0, 1, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 0, Yes, [0, 1, 3], [[0], [0, 3], [0, 1, 3]], Yes)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1, 3], [[0], [0, 3], [0, 1, 3]], Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
10. Point Guard and Every Open Neighborhood Witness
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. Point x=-1, candidate N=[2, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, -1, No, [2, 3], not-evaluated, not-evaluated)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [2, 3], not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
11. Point Guard and Every Open Neighborhood Witness
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. Point x=2, candidate N=[0, 2, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 2, Yes, [0, 2, 3], [[2], [2, 3], [0, 2, 3]], Yes)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 2, 3], [[2], [2, 3], [0, 2, 3]], Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
12. Point Guard and Every Open Neighborhood Witness
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. Point x=3, candidate N=[1, 2, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 3, Yes, [1, 2, 3], [[3], [2, 3]], Yes)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 2, 3], [[3], [2, 3]], Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
13. Point Guard and Every Open Neighborhood Witness
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. Point x=0, candidate N=[0, 1, 2, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 0, Yes, [0, 1, 2, 3], [[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], Yes)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1, 2, 3], [[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
14. Point Guard and Every Open Neighborhood Witness
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. Point x=1, candidate N=[]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 1, Yes, [], [], No)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [], [], No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
15. Point Guard and Every Open Neighborhood Witness
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. Point x=-1, candidate N=[0]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, -1, No, [0], not-evaluated, not-evaluated)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0], not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
16. Point Guard and Every Open Neighborhood Witness
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. Point x=3, candidate N=[1]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 3, Yes, [1], [], No)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1], [], No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
17. Point Guard and Every Open Neighborhood Witness
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. Point x=0, candidate N=[0, 1]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 0, Yes, [0, 1], [[0]], Yes)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1], [[0]], Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
18. Point Guard and Every Open Neighborhood Witness
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. Point x=1, candidate N=[2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 1, Yes, [2], [], No)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [2], [], No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
19. Point Guard and Every Open Neighborhood Witness
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. Point x=2, candidate N=[0, 2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 2, Yes, [0, 2], [[2]], Yes)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 2], [[2]], Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
20. Point Guard and Every Open Neighborhood Witness
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. Point x=-1, candidate N=[1, 2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, -1, No, [1, 2], not-evaluated, not-evaluated)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [1, 2], not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
21. Point Guard and Every Open Neighborhood Witness
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. Point x=0, candidate N=[0, 1, 2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 0, Yes, [0, 1, 2], [[0]], Yes)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1, 2], [[0]], Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
22. Point Guard and Every Open Neighborhood Witness
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. Point x=1, candidate N=[3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 1, Yes, [3], [], No)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [3], [], No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
23. Point Guard and Every Open Neighborhood Witness
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. Point x=2, candidate N=[0, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 2, Yes, [0, 3], [], No)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 3], [], No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
24. Point Guard and Every Open Neighborhood Witness
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. Point x=3, candidate N=[1, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(4, 3, Yes, [1, 3], [[3]], Yes)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 3], [[3]], Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
25. Point Guard and Every Open Neighborhood Witness
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. Point x=-1, candidate N=[0, 1]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.
View answer and explanation
(3, -1, No, [0, 1], not-evaluated, not-evaluated)
Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, -1, No, [0, 1], not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-witness · candidate
26. Containment, Openness, and Neighborhood Status Are Different
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. Point x=0, candidate N=[0, 1]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 0, Yes, [0, 1], Yes, Yes, Yes)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
27. Containment, Openness, and Neighborhood Status Are Different
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. Point x=1, candidate N=[2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 1, Yes, [2], No, No, No)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [2], No, No, No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
28. Containment, Openness, and Neighborhood Status Are Different
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. Point x=2, candidate N=[0, 2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 2, Yes, [0, 2], Yes, Yes, Yes)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 2], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
29. Containment, Openness, and Neighborhood Status Are Different
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. Point x=3, candidate N=[1, 2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 3, Yes, [1, 2], No, No, No)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 2], No, No, No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
30. Containment, Openness, and Neighborhood Status Are Different
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. Point x=-1, candidate N=[0, 1, 2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, -1, No, [0, 1, 2], not-evaluated, not-evaluated, not-evaluated)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0, 1, 2], not-evaluated, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
31. Containment, Openness, and Neighborhood Status Are Different
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. Point x=1, candidate N=[3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 1, Yes, [3], No, No, No)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [3], No, No, No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
32. Containment, Openness, and Neighborhood Status Are Different
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. Point x=2, candidate N=[0, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 2, Yes, [0, 3], No, No, No)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 3], No, No, No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
33. Containment, Openness, and Neighborhood Status Are Different
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. Point x=3, candidate N=[1, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 3, Yes, [1, 3], Yes, Yes, Yes)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
34. Containment, Openness, and Neighborhood Status Are Different
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. Point x=0, candidate N=[0, 1, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 0, Yes, [0, 1, 3], Yes, Yes, Yes)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
35. Containment, Openness, and Neighborhood Status Are Different
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. Point x=-1, candidate N=[2, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, -1, No, [2, 3], not-evaluated, not-evaluated, not-evaluated)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [2, 3], not-evaluated, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
36. Containment, Openness, and Neighborhood Status Are Different
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. Point x=2, candidate N=[0, 2, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 2, Yes, [0, 2, 3], Yes, Yes, Yes)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
37. Containment, Openness, and Neighborhood Status Are Different
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. Point x=3, candidate N=[1, 2, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 3, Yes, [1, 2, 3], Yes, No, Yes)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 2, 3], Yes, No, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
38. Containment, Openness, and Neighborhood Status Are Different
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. Point x=0, candidate N=[0, 1, 2, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 0, Yes, [0, 1, 2, 3], Yes, Yes, Yes)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
39. Containment, Openness, and Neighborhood Status Are Different
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. Point x=1, candidate N=[]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 1, Yes, [], No, Yes, No)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [], No, Yes, No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
40. Containment, Openness, and Neighborhood Status Are Different
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. Point x=-1, candidate N=[0]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, -1, No, [0], not-evaluated, not-evaluated, not-evaluated)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0], not-evaluated, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
41. Containment, Openness, and Neighborhood Status Are Different
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. Point x=3, candidate N=[1]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 3, Yes, [1], No, No, No)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1], No, No, No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
42. Containment, Openness, and Neighborhood Status Are Different
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. Point x=0, candidate N=[0, 1]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 0, Yes, [0, 1], Yes, No, Yes)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1], Yes, No, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
43. Containment, Openness, and Neighborhood Status Are Different
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. Point x=1, candidate N=[2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 1, Yes, [2], No, No, No)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [2], No, No, No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
44. Containment, Openness, and Neighborhood Status Are Different
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. Point x=2, candidate N=[0, 2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 2, Yes, [0, 2], Yes, No, Yes)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 2], Yes, No, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
45. Containment, Openness, and Neighborhood Status Are Different
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. Point x=-1, candidate N=[1, 2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, -1, No, [1, 2], not-evaluated, not-evaluated, not-evaluated)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [1, 2], not-evaluated, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
46. Containment, Openness, and Neighborhood Status Are Different
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. Point x=0, candidate N=[0, 1, 2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 0, Yes, [0, 1, 2], Yes, No, Yes)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1, 2], Yes, No, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
47. Containment, Openness, and Neighborhood Status Are Different
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. Point x=1, candidate N=[3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 1, Yes, [3], No, No, No)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [3], No, No, No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
48. Containment, Openness, and Neighborhood Status Are Different
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. Point x=2, candidate N=[0, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 2, Yes, [0, 3], No, No, No)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 3], No, No, No).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
49. Containment, Openness, and Neighborhood Status Are Different
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. Point x=3, candidate N=[1, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(4, 3, Yes, [1, 3], Yes, No, Yes)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 3], Yes, No, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
50. Containment, Openness, and Neighborhood Status Are Different
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. Point x=-1, candidate N=[0, 1]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.
View answer and explanation
(3, -1, No, [0, 1], not-evaluated, not-evaluated, not-evaluated)
Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, -1, No, [0, 1], not-evaluated, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-qualification · candidate
51. Neighborhood Upward-Closure Guards
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. x=0, small=[0], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
52. Neighborhood Upward-Closure Guards
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. x=1, small=[1], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
53. Neighborhood Upward-Closure Guards
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. x=2, small=[2], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
54. Neighborhood Upward-Closure Guards
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. x=3, small=[3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
55. Neighborhood Upward-Closure Guards
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. x=-1, small=[0], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, -1, No, [0], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
56. Neighborhood Upward-Closure Guards
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. x=1, small=[1], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
57. Neighborhood Upward-Closure Guards
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. x=2, small=[2], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
58. Neighborhood Upward-Closure Guards
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. x=3, small=[3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
59. Neighborhood Upward-Closure Guards
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. x=0, small=[0], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
60. Neighborhood Upward-Closure Guards
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. x=-1, small=[0, 1, 3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, -1, No, [0, 1, 3], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0, 1, 3], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
61. Neighborhood Upward-Closure Guards
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. x=2, small=[2], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
62. Neighborhood Upward-Closure Guards
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. x=3, small=[3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
63. Neighborhood Upward-Closure Guards
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. x=0, small=[0], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
64. Neighborhood Upward-Closure Guards
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. x=1, small=[1], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
65. Neighborhood Upward-Closure Guards
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. x=-1, small=[0, 2], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, -1, No, [0, 2], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0, 2], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
66. Neighborhood Upward-Closure Guards
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. x=3, small=[3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
67. Neighborhood Upward-Closure Guards
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. x=0, small=[0], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
68. Neighborhood Upward-Closure Guards
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. x=1, small=[1], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
69. Neighborhood Upward-Closure Guards
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. x=2, small=[2], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
70. Neighborhood Upward-Closure Guards
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. x=-1, small=[0, 1, 2, 3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, -1, No, [0, 1, 2, 3], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0, 1, 2, 3], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
71. Neighborhood Upward-Closure Guards
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. x=0, small=[0], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
72. Neighborhood Upward-Closure Guards
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. x=1, small=[1], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
73. Neighborhood Upward-Closure Guards
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. x=2, small=[2], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
74. Neighborhood Upward-Closure Guards
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. x=3, small=[3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
75. Neighborhood Upward-Closure Guards
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. x=-1, small=[0], large=[0, 1, 2]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
View answer and explanation
(3, -1, No, [0], [0, 1, 2], Yes, not-evaluated, not-evaluated)
Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, -1, No, [0], [0, 1, 2], Yes, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-upward-closure · candidate
76. All Neighborhoods of a Qualified Finite Point
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. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 0, [[0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
77. All Neighborhoods of a Qualified Finite Point
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. x=1 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 1, [[1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, [[1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
78. All Neighborhoods of a Qualified Finite Point
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. x=2 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 2, [[2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, [[2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
79. All Neighborhoods of a Qualified Finite Point
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. x=3 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 3, [[3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, [[3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
80. All Neighborhoods of a Qualified Finite Point
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. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 0, [[0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
81. All Neighborhoods of a Qualified Finite Point
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. x=1 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 1, [[1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, [[1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
82. All Neighborhoods of a Qualified Finite Point
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. x=2 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 2, [[2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, [[2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
83. All Neighborhoods of a Qualified Finite Point
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. x=3 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 3, [[3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, [[3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
84. All Neighborhoods of a Qualified Finite Point
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. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 0, [[0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
85. All Neighborhoods of a Qualified Finite Point
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. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 0, [[0, 1, 3], [0, 1, 2, 3]], [[0, 1, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0, 1, 3], [0, 1, 2, 3]], [[0, 1, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
86. All Neighborhoods of a Qualified Finite Point
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. x=2 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 2, [[2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, [[2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
87. All Neighborhoods of a Qualified Finite Point
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. x=3 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 3, [[3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, [[3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
88. All Neighborhoods of a Qualified Finite Point
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. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 0, [[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
89. All Neighborhoods of a Qualified Finite Point
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. x=1 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 1, [[1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, [[1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
90. All Neighborhoods of a Qualified Finite Point
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. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 0, [[0, 2], [0, 2, 3], [0, 1, 2, 3]], [[0, 2], [0, 1, 2], [0, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. 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]], [[0, 2], [0, 1, 2], [0, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
91. All Neighborhoods of a Qualified Finite Point
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. x=3 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 3, [[3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, [[3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
92. All Neighborhoods of a Qualified Finite Point
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. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 0, [[0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
93. All Neighborhoods of a Qualified Finite Point
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. x=1 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 1, [[1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, [[1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
94. All Neighborhoods of a Qualified Finite Point
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. x=2 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 2, [[2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, [[2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
95. All Neighborhoods of a Qualified Finite Point
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. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 0, [[0, 1, 2, 3]], [[0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0, 1, 2, 3]], [[0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
96. All Neighborhoods of a Qualified Finite Point
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. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 0, [[0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
97. All Neighborhoods of a Qualified Finite Point
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. x=1 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 1, [[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, [[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
98. All Neighborhoods of a Qualified Finite Point
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. x=2 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 2, [[2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, [[2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
99. All Neighborhoods of a Qualified Finite Point
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. x=3 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(4, 3, [[3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, [[3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
100. All Neighborhoods of a Qualified Finite Point
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. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.
View answer and explanation
(3, 0, [[0], [0, 1], [0, 1, 2]], [[0], [0, 1], [0, 2], [0, 1, 2]])
The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 0, [[0], [0, 1], [0, 1, 2]], [[0], [0, 1], [0, 2], [0, 1, 2]]).
Reviewed step-by-step derivation
Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate
Canonical question JSONL · English JSONL · Validation report