# Carrier Coverage, Local Refinement and Complete Topology Generation

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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-20261107-r03-topological-basis-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

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

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



## 1. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0001/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 2. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0002/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 3. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0003/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 4. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0004/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 5. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0005/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 6. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0006/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 7. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0007/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 8. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0008/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 9. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0009/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 10. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0010/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 11. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0011/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 12. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0012/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 13. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0013/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 14. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0014/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 15. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0015/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 16. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0016/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 17. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0017/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 18. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0018/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 19. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0019/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 20. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0020/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 21. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0021/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 22. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0022/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 23. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0023/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 24. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0024/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 25. Coverage and Pointwise Intersection Refinement Are Separate Axioms

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [2, 3], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).

View answer and explanation(Yes, Yes, null, Yes)

Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0025/)

Topological basis · qf-mod38-topbasis-basis-qualification · candidate





## 26. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0026/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 27. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0027/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 28. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0028/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 29. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0029/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 30. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0030/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 31. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0031/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 32. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0032/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 33. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0033/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 34. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0034/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 35. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0035/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 36. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0036/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 37. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0037/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 38. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0038/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 39. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0039/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 40. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0040/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 41. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0041/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 42. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0042/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 43. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0043/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 44. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0044/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 45. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0045/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 46. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0046/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 47. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0047/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 48. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0048/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 49. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0049/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 50. Generate All Unions Only after the Basis Axioms Pass

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [2, 3], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).

View answer and explanation([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)

Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0050/)

Topological basis · qf-mod38-topbasis-generated-topology · candidate





## 51. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0051/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 52. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0052/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 53. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0053/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 54. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0054/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 55. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0055/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 56. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0056/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 57. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0057/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 58. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0058/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 59. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0059/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 60. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0060/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 61. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0061/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 62. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0062/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 63. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0063/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 64. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0064/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 65. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0065/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 66. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0066/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 67. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0067/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 68. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0068/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 69. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0069/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 70. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0070/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 71. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0071/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 72. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0072/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 73. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0073/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 74. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0074/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 75. A Basis for Some Topology Need Not Generate This Given Topology

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [2, 3], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).

View answer and explanation(Yes, Yes, Yes, Yes, Yes)

Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0075/)

Topological basis · qf-mod38-topbasis-existing-topology-audit · candidate





## 76. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0076/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 77. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0077/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 78. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0078/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 79. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0079/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 80. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0080/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 81. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0081/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 82. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0082/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 83. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0083/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 84. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 3], [1, 2], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0084/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 85. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0085/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 86. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0086/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 87. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0087/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 88. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0088/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 89. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0089/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 90. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0090/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 91. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0091/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 92. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [1, 4], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0092/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 93. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [1, 4], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0093/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 94. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0094/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 95. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0095/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 96. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [1, 4], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0096/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 97. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [1, 4], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0097/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 98. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0098/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 99. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0099/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate





## 100. A Cover Alone Does Not Supply the Local Basis Axiom

Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).

The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status.

View answer and explanation(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)

Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r03-topological-basis-0100/)

Topological basis · qf-mod38-topbasis-subbasis-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261107-r03-topological-basis-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261107-r03-topological-basis-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261107-r03-topological-basis-foundations/validation-report.json)