Carrier Coverage, Local Refinement and Complete Topology Generation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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).

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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.

Reviewed step-by-step derivation

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