# Both Continuity Directions, Finite Discrete Scope and Metric Boundaries

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261105-r05-homeomorphism-foundations · 2026-10-14 · 100 questions

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

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

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



## 1. 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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0001/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0002/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0003/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0004/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0005/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0006/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0007/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0008/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0009/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0010/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0011/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0012/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0013/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0014/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0015/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0016/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0017/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0018/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0019/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0020/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0021/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0022/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0023/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0024/)

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).

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

Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0025/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0026/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0027/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0028/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0029/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0030/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0031/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0032/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0033/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0034/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0035/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0036/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0037/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0038/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0039/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0040/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0041/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0042/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0043/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0044/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0045/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0046/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0047/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0048/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0049/)

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).

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

Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0050/)

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).

View answer and explanation(5, [0, 1, 2, 3, 4], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 1, 2, 3, 4], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0051/)

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).

View answer and explanation(5, [0, 1, 2, 4, 3], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 1, 2, 4, 3], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0052/)

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).

View answer and explanation(5, [0, 1, 3, 2, 4], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 1, 3, 2, 4], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0053/)

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).

View answer and explanation(5, [0, 1, 3, 4, 2], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 1, 3, 4, 2], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0054/)

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).

View answer and explanation(5, [0, 1, 4, 2, 3], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 1, 4, 2, 3], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0055/)

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).

View answer and explanation(5, [0, 1, 4, 3, 2], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 1, 4, 3, 2], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0056/)

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).

View answer and explanation(5, [0, 2, 1, 3, 4], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 2, 1, 3, 4], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0057/)

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).

View answer and explanation(5, [0, 2, 1, 4, 3], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 2, 1, 4, 3], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0058/)

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).

View answer and explanation(5, [0, 2, 3, 1, 4], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 2, 3, 1, 4], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0059/)

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).

View answer and explanation(5, [0, 2, 3, 4, 1], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 2, 3, 4, 1], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0060/)

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).

View answer and explanation(5, [0, 2, 4, 1, 3], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 2, 4, 1, 3], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0061/)

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).

View answer and explanation(5, [0, 2, 4, 3, 1], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 2, 4, 3, 1], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0062/)

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).

View answer and explanation(5, [0, 3, 1, 2, 4], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 3, 1, 2, 4], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0063/)

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).

View answer and explanation(5, [0, 3, 1, 4, 2], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 3, 1, 4, 2], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0064/)

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).

View answer and explanation(5, [0, 3, 2, 1, 4], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 3, 2, 1, 4], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0065/)

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).

View answer and explanation(5, [0, 3, 2, 4, 1], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 3, 2, 4, 1], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0066/)

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).

View answer and explanation(5, [0, 3, 4, 1, 2], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 3, 4, 1, 2], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0067/)

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).

View answer and explanation(5, [0, 3, 4, 2, 1], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 3, 4, 2, 1], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0068/)

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).

View answer and explanation(5, [0, 4, 1, 2, 3], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 4, 1, 2, 3], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0069/)

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).

View answer and explanation(5, [0, 4, 1, 3, 2], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 4, 1, 3, 2], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0070/)

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).

View answer and explanation(5, [0, 4, 2, 1, 3], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 4, 2, 1, 3], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0071/)

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).

View answer and explanation(5, [0, 4, 2, 3, 1], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 4, 2, 3, 1], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0072/)

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).

View answer and explanation(5, [0, 4, 3, 1, 2], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 4, 3, 1, 2], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0073/)

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).

View answer and explanation(5, [0, 4, 3, 2, 1], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 4, 3, 2, 1], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0074/)

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).

View answer and explanation(5, [1, 0, 2, 3, 4], Yes, Yes, Yes, finite_discrete_scope_only)

Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [1, 0, 2, 3, 4], Yes, Yes, Yes, finite_discrete_scope_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0075/)

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.

View answer and explanation(5, Yes, No, 1, 2, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 1, 2, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0076/)

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.

View answer and explanation(5, Yes, No, 2, 3, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 2, 3, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0077/)

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.

View answer and explanation(5, Yes, No, 3, 4, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 3, 4, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0078/)

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.

View answer and explanation(5, Yes, No, 4, 5, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 4, 5, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0079/)

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.

View answer and explanation(5, Yes, No, 5, 6, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 5, 6, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0080/)

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.

View answer and explanation(5, Yes, No, 6, 7, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 6, 7, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0081/)

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.

View answer and explanation(5, Yes, No, 7, 8, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 7, 8, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0082/)

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.

View answer and explanation(5, Yes, No, 8, 9, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 8, 9, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0083/)

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.

View answer and explanation(5, Yes, No, 9, 10, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 9, 10, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0084/)

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.

View answer and explanation(5, Yes, No, 10, 11, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 10, 11, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0085/)

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.

View answer and explanation(5, Yes, No, 11, 12, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 11, 12, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0086/)

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.

View answer and explanation(5, Yes, No, 12, 13, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 12, 13, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0087/)

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.

View answer and explanation(5, Yes, No, 13, 14, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 13, 14, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0088/)

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.

View answer and explanation(5, Yes, No, 14, 15, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 14, 15, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0089/)

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.

View answer and explanation(5, Yes, No, 15, 16, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 15, 16, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0090/)

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.

View answer and explanation(5, Yes, No, 16, 17, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 16, 17, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0091/)

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.

View answer and explanation(5, Yes, No, 17, 18, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 17, 18, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0092/)

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.

View answer and explanation(5, Yes, No, 18, 19, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 18, 19, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0093/)

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.

View answer and explanation(5, Yes, No, 19, 20, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 19, 20, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0094/)

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.

View answer and explanation(5, Yes, No, 20, 21, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 20, 21, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0095/)

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.

View answer and explanation(5, Yes, No, 21, 22, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 21, 22, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0096/)

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.

View answer and explanation(5, Yes, No, 22, 23, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 22, 23, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0097/)

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.

View answer and explanation(5, Yes, No, 23, 24, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 23, 24, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0098/)

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.

View answer and explanation(5, Yes, No, 24, 25, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 24, 25, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0099/)

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.

View answer and explanation(5, Yes, No, 25, 26, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)

The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 25, 26, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r05-homeomorphism-0100/)

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



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261105-r05-homeomorphism-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261105-r05-homeomorphism-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261105-r05-homeomorphism-foundations/validation-report.json)