# Point-Qualified Neighborhoods and All Open Witnesses

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

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

batch-20261013-advance-20261019-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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r02-neighborhood-0100/)

Neighborhood · qf-ext19-neighborhood-complete-inventory · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261019-r02-neighborhood-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261019-r02-neighborhood-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261019-r02-neighborhood-foundations/validation-report.json)