# Relative Open Covers, Genuine Subcovers and Finite Compactness Boundaries

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-20261109-r03-compact-subset-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.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0001/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 2. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0002/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 3. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0003/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 4. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0004/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 5. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0005/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 6. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0006/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 7. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0007/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 8. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0008/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 9. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0009/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 10. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0010/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 11. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0011/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 12. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0012/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 13. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0013/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 14. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0014/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 15. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0015/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 16. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0016/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 17. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0017/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 18. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0018/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 19. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0019/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 20. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0020/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 21. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0021/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 22. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0022/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 23. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0023/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 24. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0024/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 25. The Complete Finite Relative Topology Supplies Compactness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], subset K=[0, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).

View answer and explanation([0, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)

Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([0, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0025/)

Compact subset · qf-mod40-compact-finite-compactness · candidate





## 26. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1], [1, 3], [1, 3, 4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1], [1, 3], [1, 3, 4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1], [1, 3], [1, 3, 4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0026/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 27. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1], [1, 3, 4], [1, 4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1], [1, 3, 4], [1, 4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1], [1, 3, 4], [1, 4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0027/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 28. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1], [1, 2], [1, 2, 4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1], [1, 2], [1, 2, 4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1], [1, 2], [1, 2, 4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0028/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 29. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1], [1, 2, 4], [1, 4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1], [1, 2, 4], [1, 4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1], [1, 2, 4], [1, 4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0029/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 30. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1], [1, 2], [1, 2, 3]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1], [1, 2], [1, 2, 3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1], [1, 2], [1, 2, 3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0030/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 31. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1], [1, 2, 3], [1, 3]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1], [1, 2, 3], [1, 3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1], [1, 2, 3], [1, 3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0031/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 32. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [2], [2, 3], [2, 3, 4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [2], [2, 3], [2, 3, 4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [2], [2, 3], [2, 3, 4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0032/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 33. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [2], [2, 3, 4], [2, 4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [2], [2, 3, 4], [2, 4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [2], [2, 3, 4], [2, 4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0033/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 34. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2], [1, 2, 4], [2]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1, 2], [1, 2, 4], [2]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2], [1, 2, 4], [2]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0034/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 35. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2, 4], [2], [2, 4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1, 2, 4], [2], [2, 4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2, 4], [2], [2, 4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0035/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 36. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2], [1, 2, 3], [2]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1, 2], [1, 2, 3], [2]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2], [1, 2, 3], [2]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0036/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 37. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2, 3], [2], [2, 3]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1, 2, 3], [2], [2, 3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2, 3], [2], [2, 3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0037/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 38. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [2, 3], [2, 3, 4], [3]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [2, 3], [2, 3, 4], [3]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [2, 3], [2, 3, 4], [3]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0038/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 39. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [2, 3, 4], [3], [3, 4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [2, 3, 4], [3], [3, 4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [2, 3, 4], [3], [3, 4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0039/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 40. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 3], [1, 3, 4], [3]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1, 3], [1, 3, 4], [3]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 3], [1, 3, 4], [3]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0040/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 41. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 3, 4], [3], [3, 4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1, 3, 4], [3], [3, 4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 3, 4], [3], [3, 4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0041/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 42. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2, 3], [1, 3], [3]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1, 2, 3], [1, 3], [3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2, 3], [1, 3], [3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0042/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 43. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2, 3], [2, 3], [3]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1, 2, 3], [2, 3], [3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2, 3], [2, 3], [3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0043/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 44. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [2, 3, 4], [2, 4], [4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [2, 3, 4], [2, 4], [4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [2, 3, 4], [2, 4], [4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0044/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 45. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [2, 3, 4], [3, 4], [4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [2, 3, 4], [3, 4], [4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [2, 3, 4], [3, 4], [4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0045/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 46. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 3, 4], [1, 4], [4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1, 3, 4], [1, 4], [4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 3, 4], [1, 4], [4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0046/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 47. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 3, 4], [3, 4], [4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1, 3, 4], [3, 4], [4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 3, 4], [3, 4], [4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0047/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 48. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2, 4], [1, 4], [4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1, 2, 4], [1, 4], [4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2, 4], [1, 4], [4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0048/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 49. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2, 4], [2, 4], [4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [1, 2, 4], [2, 4], [4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2, 4], [2, 4], [4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0049/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 50. Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], subset K=[0, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [0], [0, 3], [0, 3, 4]]. Give (cover, covers_subset, union, boundary).

View answer and explanation([[], [0], [0, 3], [0, 3, 4]], Yes, [0, 3, 4], single_cover_audit_not_general_compactness_proof)

Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [0], [0, 3], [0, 3, 4]], Yes, [0, 3, 4], single_cover_audit_not_general_compactness_proof).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0050/)

Compact subset · qf-mod40-compact-open-cover-audit · candidate





## 51. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1], [1, 3], [1, 3, 4]], selected indices=[3]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([3], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([3], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0051/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 52. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1], [1, 3, 4], [1, 4]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([2], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0052/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 53. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1], [1, 2], [1, 2, 4]], selected indices=[3]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([3], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([3], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0053/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 54. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1], [1, 2, 4], [1, 4]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([2], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0054/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 55. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1], [1, 2], [1, 2, 3]], selected indices=[3]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([3], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([3], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0055/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 56. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1], [1, 2, 3], [1, 3]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([2], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0056/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 57. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [2], [2, 3], [2, 3, 4]], selected indices=[3]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([3], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([3], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0057/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 58. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [2], [2, 3, 4], [2, 4]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([2], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0058/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 59. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2], [1, 2, 4], [2]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([2], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0059/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 60. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2, 4], [2], [2, 4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([1], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0060/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 61. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2], [1, 2, 3], [2]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([2], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0061/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 62. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2, 3], [2], [2, 3]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([1], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0062/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 63. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [2, 3], [2, 3, 4], [3]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([2], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0063/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 64. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [2, 3, 4], [3], [3, 4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([1], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0064/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 65. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 3], [1, 3, 4], [3]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([2], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0065/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 66. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 3, 4], [3], [3, 4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([1], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0066/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 67. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2, 3], [1, 3], [3]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([1], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0067/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 68. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2, 3], [2, 3], [3]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([1], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0068/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 69. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [2, 3, 4], [2, 4], [4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([1], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0069/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 70. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [2, 3, 4], [3, 4], [4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([1], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0070/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 71. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 3, 4], [1, 4], [4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([1], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0071/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 72. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 3, 4], [3, 4], [4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([1], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0072/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 73. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2, 4], [1, 4], [4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([1], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0073/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 74. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2, 4], [2, 4], [4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([1], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0074/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 75. Original Cover Membership and Actual Coverage Are Separate Checks

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], subset K=[0, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [0], [0, 3], [0, 3, 4]], selected indices=[3]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).

covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made.

View answer and explanation([3], [[0, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)

Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([3], [[0, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0075/)

Compact subset · qf-mod40-compact-subcover-certificate · candidate





## 76. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0076/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 77. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0077/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 78. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0078/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 79. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0079/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 80. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0080/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 81. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0081/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 82. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0082/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 83. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0083/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 84. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0084/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 85. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0085/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 86. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0086/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 87. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0087/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 88. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0088/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 89. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0089/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 90. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0090/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 91. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0091/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 92. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0092/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 93. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0093/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 94. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0094/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 95. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0095/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 96. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0096/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 97. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0097/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 98. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0098/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 99. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0099/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate





## 100. Compactness Does Not Supply General Ambient Closedness or Boundedness

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], subset K=[0, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).

The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric.

View answer and explanation([0, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([0, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261109-r03-compact-subset-0100/)

Compact subset · qf-mod40-compact-nonproperty-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261109-r03-compact-subset-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261109-r03-compact-subset-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261109-r03-compact-subset-foundations/validation-report.json)