# 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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r02-closure-and-interior-0100/)

Topological closure and interior · qf-mod32-topology-interior-membership · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261101-r02-closure-and-interior-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261101-r02-closure-and-interior-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261101-r02-closure-and-interior-foundations/validation-report.json)