Relative Open Covers, Genuine Subcovers and Finite Compactness Boundaries
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Compact subset · qf-mod40-compact-nonproperty-boundary · candidate
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