Complete Topology-Relative Closure, Interior and Membership
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-20261014-advance-20261101-r02-closure-and-interior-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
2. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
3. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
4. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
5. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
6. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
7. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
8. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 2], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
9. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
10. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
11. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
12. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
13. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
14. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
15. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
16. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 3], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
17. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
18. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
19. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
20. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([1, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
21. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
22. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([2, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
23. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
24. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([3, 4], [[0, 1, 2, 3, 4], [1, 2, 3, 4]], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
25. Closure Intersects All Closed Supersets in the Specified Topology
Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[0, 2]. Give (subset,closed_supersets,closure). Closed candidates are complements of the supplied opens in their input order; retain that order among qualifying supersets.
View answer and explanation
([0, 2], [[0, 1, 2, 3, 4], [0, 2, 3, 4]], [0, 2, 3, 4])
Form every closed set relative to this carrier, retain all that contain the subset and intersect them. One selected closed superset is not generally the closure. The exact result, retaining the requested order, is ([0, 2], [[0, 1, 2, 3, 4], [0, 2, 3, 4]], [0, 2, 3, 4]).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure · candidate
26. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([1, 2], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 2], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
27. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([1, 2], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 2], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
28. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([1, 3], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 3], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
29. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([1, 3], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 3], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
30. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([1, 4], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 4], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
31. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([1, 4], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 4], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
32. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([1, 2], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 2], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
33. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([1, 2], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 2], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
34. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([2, 3], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 3], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
35. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([2, 3], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 3], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
36. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([2, 4], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 4], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
37. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([2, 4], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 4], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
38. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([1, 3], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 3], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
39. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([1, 3], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 3], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
40. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([2, 3], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 3], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
41. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([2, 3], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 3], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
42. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([3, 4], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([3, 4], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
43. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([3, 4], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([3, 4], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
44. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([1, 4], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 4], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
45. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([1, 4], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([1, 4], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
46. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([2, 4], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 4], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
47. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([2, 4], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([2, 4], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
48. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([3, 4], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([3, 4], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
49. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([3, 4], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([3, 4], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
50. Interior Unites All Open Subsets, Not One Convenient Open Set
Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[0, 2]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.
View answer and explanation
([0, 2], [[]], [])
Select every open contained in the subset and take their union in increasing carrier order. The result is relative to this exact topology, not an unstated metric. The exact result, retaining the requested order, is ([0, 2], [[]], []).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior · candidate
51. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=0. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(0, [1, 2, 3, 4], No, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
52. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=1. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(1, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
53. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=3. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(3, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
54. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=4. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(4, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
55. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=3. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(3, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
56. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=0. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(0, [1, 2, 3, 4], No, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
57. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=2. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(2, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
58. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=1. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(1, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
59. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=1. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(1, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
60. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=1. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(1, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
61. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=0. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(0, [1, 2, 3, 4], No, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
62. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=2. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(2, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
63. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=1. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(1, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
64. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=4. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(4, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
65. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=4. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(4, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
66. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=0. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(0, [1, 2, 3, 4], No, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
67. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=3. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(3, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
68. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=4. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(4, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
69. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=2. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(2, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
70. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=2. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(2, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
71. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=0. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(0, [1, 2, 3, 4], No, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
72. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=4. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(4, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
73. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=3. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(3, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
74. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=2. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(2, [1, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
75. Closure Membership May Hold outside the Original Subset
Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[0, 2], point=4. Give (point,closure,in_closure?,topology_relative_closure).
View answer and explanation
(4, [0, 2, 3, 4], Yes, topology_relative_closure)
Compute the full intersection of closed supersets first, then test this point against that closure. Do not replace closure membership with membership in the original subset. The exact result, retaining the requested order, is (4, [0, 2, 3, 4], Yes, topology_relative_closure).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-closure-membership · candidate
76. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=0. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(0, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (0, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
77. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=1. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(1, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (1, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
78. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=3. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(3, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (3, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
79. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=4. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(4, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (4, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
80. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=3. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(3, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (3, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
81. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=0. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(0, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (0, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
82. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=2. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(2, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (2, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
83. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 2], point=1. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(1, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (1, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
84. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=1. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(1, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (1, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
85. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=1. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(1, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (1, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
86. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=0. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(0, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (0, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
87. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=2. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(2, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (2, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
88. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=1. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(1, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (1, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
89. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 3], point=4. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(4, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (4, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
90. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=4. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(4, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (4, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
91. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 3], point=0. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(0, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (0, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
92. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=3. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(3, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (3, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
93. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=4. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(4, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (4, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
94. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=2. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(2, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (2, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
95. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[1, 4], point=2. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(2, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (2, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
96. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=0. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(0, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (0, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
97. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[2, 4], point=4. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(4, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (4, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
98. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=3. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(3, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (3, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
99. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=2. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(2, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (2, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
100. Interior Membership Needs an Open Neighborhood inside the Subset
Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[0, 2], point=4. Give (point,interior,in_interior?,topology_relative_interior).
View answer and explanation
(4, [], No, topology_relative_interior)
Unite all opens contained in the subset, then test the point. Membership in the subset alone need not give membership in its interior. The exact result, retaining the requested order, is (4, [], No, topology_relative_interior).
Reviewed step-by-step derivation
Topological closure and interior · qf-mod32-topology-interior-membership · candidate
Canonical question JSONL · English JSONL · Validation report