Both Continuity Directions, Finite Discrete Scope and Metric Boundaries

1. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 1, 2, 3, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

2. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 1, 2, 4, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

3. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 1, 3, 2, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

4. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 1, 3, 4, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

5. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 1, 4, 2, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

6. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 1, 4, 3, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

7. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 2, 1, 3, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

8. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 2, 1, 4, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

9. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 2, 3, 1, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

10. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 2, 3, 4, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

11. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 2, 4, 1, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

12. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 2, 4, 3, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

13. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 3, 1, 2, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

14. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 3, 1, 4, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

15. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 3, 2, 1, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

16. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 3, 2, 4, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

17. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 3, 4, 1, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

18. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 3, 4, 2, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

19. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 4, 1, 2, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

20. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 4, 1, 3, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

21. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 4, 2, 1, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

22. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 4, 2, 3, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

23. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 4, 3, 1, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

24. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 4, 3, 2, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

25. A Homeomorphism Has Both Forward and Inverse Continuity

Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0, 1, 2], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [1], [1, 3, 4], [3, 4]] and total candidate bijection map=[1, 0, 2, 3, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-topology-audit · candidate

26. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 1, 2, 3, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

27. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 1, 2, 4, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

28. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 1, 3, 2, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

29. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 1, 3, 4, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

30. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 1, 4, 2, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

31. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 1, 4, 3, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

32. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 2, 1, 3, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

33. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 2, 1, 4, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

34. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 2, 3, 1, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

35. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 2, 3, 4, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

36. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 2, 4, 1, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

37. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 2, 4, 3, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

38. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 3, 1, 2, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

39. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 3, 1, 4, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

40. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 3, 2, 1, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

41. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 3, 2, 4, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

42. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 3, 4, 1, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

43. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 3, 4, 2, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

44. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 4, 1, 2, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

45. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 4, 1, 3, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

46. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 4, 2, 1, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

47. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 4, 2, 3, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

48. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 4, 3, 1, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

49. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 4, 3, 2, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

50. A Bijection Must Still Pass Both Continuity Directions

Use source and target on Fin(5)={0,…,4}, with complete source_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]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[1, 0, 2, 3, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-continuous-bijection-boundary · candidate

51. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 1, 2, 3, 4]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

52. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 1, 2, 4, 3]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

53. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 1, 3, 2, 4]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

54. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 1, 3, 4, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

55. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 1, 4, 2, 3]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

56. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 1, 4, 3, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

57. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 2, 1, 3, 4]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

58. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 2, 1, 4, 3]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

59. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 2, 3, 1, 4]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

60. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 2, 3, 4, 1]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

61. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 2, 4, 1, 3]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

62. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 2, 4, 3, 1]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

63. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 3, 1, 2, 4]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

64. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 3, 1, 4, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

65. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 3, 2, 1, 4]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

66. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 3, 2, 4, 1]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

67. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 3, 4, 1, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

68. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 3, 4, 2, 1]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

69. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 4, 1, 2, 3]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

70. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 4, 1, 3, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

71. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 4, 2, 1, 3]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

72. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 4, 2, 3, 1]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

73. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 4, 3, 1, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

74. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 4, 3, 2, 1]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

75. A Finite Discrete Bijection Is a Homeomorphism within That Scope

Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[1, 0, 2, 3, 4]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-finite-discrete-bijection · candidate

76. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 1, 2, 3, 4], positive source_scale=1 and target_scale=2. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

77. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 1, 2, 4, 3], positive source_scale=2 and target_scale=3. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

78. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 1, 3, 2, 4], positive source_scale=3 and target_scale=4. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

79. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 1, 3, 4, 2], positive source_scale=4 and target_scale=5. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

80. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 1, 4, 2, 3], positive source_scale=5 and target_scale=6. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

81. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 1, 4, 3, 2], positive source_scale=6 and target_scale=7. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

82. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 2, 1, 3, 4], positive source_scale=7 and target_scale=8. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

83. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 2, 1, 4, 3], positive source_scale=8 and target_scale=9. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

84. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 2, 3, 1, 4], positive source_scale=9 and target_scale=10. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

85. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 2, 3, 4, 1], positive source_scale=10 and target_scale=11. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

86. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 2, 4, 1, 3], positive source_scale=11 and target_scale=12. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

87. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 2, 4, 3, 1], positive source_scale=12 and target_scale=13. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

88. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 3, 1, 2, 4], positive source_scale=13 and target_scale=14. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

89. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 3, 1, 4, 2], positive source_scale=14 and target_scale=15. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

90. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 3, 2, 1, 4], positive source_scale=15 and target_scale=16. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

91. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 3, 2, 4, 1], positive source_scale=16 and target_scale=17. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

92. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 3, 4, 1, 2], positive source_scale=17 and target_scale=18. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

93. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 3, 4, 2, 1], positive source_scale=18 and target_scale=19. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

94. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 4, 1, 2, 3], positive source_scale=19 and target_scale=20. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

95. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 4, 1, 3, 2], positive source_scale=20 and target_scale=21. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

96. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 4, 2, 1, 3], positive source_scale=21 and target_scale=22. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

97. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 4, 2, 3, 1], positive source_scale=22 and target_scale=23. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

98. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 4, 3, 1, 2], positive source_scale=23 and target_scale=24. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

99. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 4, 3, 2, 1], positive source_scale=24 and target_scale=25. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate

100. Equal Scale Parameters Are Not an Unqualified Isometry Test

Use discrete spaces on Fin(5)={0,…,4}, bijection map=[1, 0, 2, 3, 4], positive source_scale=25 and target_scale=26. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).

The third field compares scale parameters; its No does not disprove isometry of a singleton space.

Reviewed step-by-step derivation

Homeomorphism · qf-mod36-homeo-metric-nonimplication · candidate