# Complete Open-Preimage Continuity and Open-Map Contrasts

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-20261013-advance-20261019-r03-topological-continuity-foundations · 2026-10-13 · 100 questions

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



## 1. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 1, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [], [], Yes)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0001/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 2. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [1], [0], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1], [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0002/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 3. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 0, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [1], [2], Yes)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1], [2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0003/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 4. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[0, 2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

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

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [0, 2, 3], [0, 1, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0004/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 5. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 2, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[2] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [2], [1], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [2], [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0005/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 6. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [], [], Yes)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0006/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 7. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 1, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1, 2] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [1, 2], [0, 1], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1, 2], [0, 1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0007/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 8. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [2, 3], [0, 3], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [2, 3], [0, 3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0008/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 9. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 3, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [3], [1], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [3], [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0009/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 10. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[0, 1, 2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

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

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0010/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 11. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 3, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [1, 3], [1, 3], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1, 3], [1, 3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0011/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 12. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [3], [0], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [3], [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0012/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 13. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 1, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [2, 3], [2, 3], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [2, 3], [2, 3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0013/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 14. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1, 2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [1, 2, 3], [0, 2, 3], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1, 2, 3], [0, 2, 3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0014/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 15. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 0, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1, 2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

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

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1, 2, 3], [0, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0015/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 16. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [], [], Yes)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0016/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 17. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 2, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [], [], Yes)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0017/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 18. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[0, 1] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [0, 1], [0, 3], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [0, 1], [0, 3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0018/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 19. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 1, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [1], [1], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1], [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0019/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 20. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[0, 1, 2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

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

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0020/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 21. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 3, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[2] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [2], [2], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [2], [2], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0021/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 22. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [1], [0], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1], [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0022/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 23. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 3, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1, 2] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [1, 2], [0, 2], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1, 2], [0, 2], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0023/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 24. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1, 2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(4, 4, [1, 2, 3], [0, 1, 2], No)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1, 2, 3], [0, 1, 2], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0024/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 25. One Declared Target-Open Preimage

Use complete finite topologies X=Fin(3)={0,…,2}, opens_X=[[], [0], [0, 1], [0, 1, 2]] and Y=Fin(3)={0,…,2}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]. The complete total map has images=[0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.

View answer and explanation(3, 3, [], [], Yes)

Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 3, [], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0025/)

Continuous maps between topological spaces · qf-ext19-continuity-single-preimage · candidate





## 26. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 1, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([0, 1, 2, 3], [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0026/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 27. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([1, 0, 2, 3], [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 3], [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0027/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 28. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 0, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([2, 0, 1, 3], [[[], [], Yes], [[0], [1], No], [[1], [2], Yes], [[0, 1], [1, 2], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [1, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 3], [[[], [], Yes], [[0], [1], No], [[1], [2], Yes], [[0, 1], [1, 2], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [1, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0028/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 29. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([3, 0, 1, 2], [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 0, 1, 2], [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0029/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 30. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 2, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([0, 2, 1, 3], [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], Yes], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 3], [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], Yes], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0030/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 31. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([1, 2, 0, 3], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 3], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0031/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 32. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 1, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([2, 1, 0, 3], [[[], [], Yes], [[0], [2], Yes], [[1], [1], No], [[0, 1], [1, 2], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [2, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 3], [[[], [], Yes], [[0], [2], Yes], [[1], [1], No], [[0, 1], [1, 2], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [2, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0032/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 33. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([3, 1, 0, 2], [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1, 0, 2], [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0033/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 34. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 3, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([0, 3, 1, 2], [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], No], [[2], [3], No], [[0, 2], [0, 3], Yes], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 1, 2], [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], No], [[2], [3], No], [[0, 2], [0, 3], Yes], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0034/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 35. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([1, 3, 0, 2], [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 0, 2], [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0035/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 36. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 3, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([2, 3, 0, 1], [[[], [], Yes], [[0], [2], Yes], [[1], [3], No], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [1, 2], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 0, 1], [[[], [], Yes], [[0], [2], Yes], [[1], [3], No], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [1, 2], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0036/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 37. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([3, 2, 0, 1], [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2, 0, 1], [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0037/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 38. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 1, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([0, 1, 3, 2], [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [3], No], [[0, 2], [0, 3], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], Yes], [[3], [2], No], [[0, 3], [0, 2], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3, 2], [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [3], No], [[0, 2], [0, 3], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], Yes], [[3], [2], No], [[0, 3], [0, 2], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0038/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 39. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([1, 0, 3, 2], [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 3, 2], [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0039/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 40. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 0, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([2, 0, 3, 1], [[[], [], Yes], [[0], [1], No], [[1], [3], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [1, 2], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 2], Yes], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 3, 1], [[[], [], Yes], [[0], [1], No], [[1], [3], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [1, 2], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 2], Yes], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0040/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 41. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([3, 0, 2, 1], [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 0, 2, 1], [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0041/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 42. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 2, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([0, 2, 3, 1], [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], No], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], No], [[0, 3], [0, 2], Yes], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3, 1], [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], No], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], No], [[0, 3], [0, 2], Yes], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0042/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 43. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([1, 2, 3, 0], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 3], No], [[0, 1, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3, 0], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 3], No], [[0, 1, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0043/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 44. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 1, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([2, 1, 3, 0], [[[], [], Yes], [[0], [3], No], [[1], [1], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [2, 3], No], [[1, 3], [1, 2], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 2], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 3, 0], [[[], [], Yes], [[0], [3], No], [[1], [1], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [2, 3], No], [[1, 3], [1, 2], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 2], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0044/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 45. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([3, 1, 2, 0], [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1, 2, 0], [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0045/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 46. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 3, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([0, 3, 2, 1], [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 2, 1], [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0046/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 47. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([1, 3, 2, 0], [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 2, 0], [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0047/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 48. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 3, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([2, 3, 1, 0], [[[], [], Yes], [[0], [3], No], [[1], [2], Yes], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 1, 0], [[[], [], Yes], [[0], [3], No], [[1], [2], Yes], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0048/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 49. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([3, 2, 1, 0], [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2, 1, 0], [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0049/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 50. Every Target Open Set and Its Full Preimage

Use complete finite topologies X=Fin(3)={0,…,2}, opens_X=[[], [0], [0, 1], [0, 1, 2]] and Y=Fin(3)={0,…,2}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]. The complete total map has images=[0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.

View answer and explanation([0, 1, 2], [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes]], No)

Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2], [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0050/)

Continuous maps between topological spaces · qf-ext19-continuity-full-audit · candidate





## 51. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 1, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[0, 1, 2, 3]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([0, 1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0051/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 52. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[3, 1, 0, 2]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([1, 0, 2, 3], [3, 1, 0, 2], [1, 3, 0, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 3], [3, 1, 0, 2], [1, 3, 0, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0052/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 53. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 0, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[2, 0, 3, 1]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([2, 0, 1, 3], [2, 0, 3, 1], [3, 2, 0, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 3], [2, 0, 3, 1], [3, 2, 0, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0053/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 54. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[1, 3, 2, 0]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([3, 0, 1, 2], [1, 3, 2, 0], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 0, 1, 2], [1, 3, 2, 0], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0054/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 55. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 2, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[0, 2, 1, 3]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([0, 2, 1, 3], [0, 2, 1, 3], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 3], [0, 2, 1, 3], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0055/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 56. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[3, 2, 0, 1]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([1, 2, 0, 3], [3, 2, 0, 1], [2, 0, 3, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 3], [3, 2, 0, 1], [2, 0, 3, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0056/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 57. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 1, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[2, 1, 3, 0]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([2, 1, 0, 3], [2, 1, 3, 0], [3, 1, 2, 0], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 3], [2, 1, 3, 0], [3, 1, 2, 0], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0057/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 58. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[1, 0, 2, 3]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([3, 1, 0, 2], [1, 0, 2, 3], [3, 0, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1, 0, 2], [1, 0, 2, 3], [3, 0, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0058/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 59. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 3, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[0, 3, 1, 2]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([0, 3, 1, 2], [0, 3, 1, 2], [0, 2, 3, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 1, 2], [0, 3, 1, 2], [0, 2, 3, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0059/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 60. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[3, 0, 2, 1]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([1, 3, 0, 2], [3, 0, 2, 1], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 0, 2], [3, 0, 2, 1], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0060/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 61. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 3, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[2, 3, 1, 0]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([2, 3, 0, 1], [2, 3, 1, 0], [1, 0, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 0, 1], [2, 3, 1, 0], [1, 0, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0061/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 62. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[1, 2, 0, 3]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([3, 2, 0, 1], [1, 2, 0, 3], [3, 0, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2, 0, 1], [1, 2, 0, 3], [3, 0, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0062/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 63. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 1, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[0, 1, 3, 2]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([0, 1, 3, 2], [0, 1, 3, 2], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3, 2], [0, 1, 3, 2], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0063/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 64. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[3, 1, 2, 0]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([1, 0, 3, 2], [3, 1, 2, 0], [1, 3, 0, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 3, 2], [3, 1, 2, 0], [1, 3, 0, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0064/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 65. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 0, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[2, 0, 1, 3]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([2, 0, 3, 1], [2, 0, 1, 3], [1, 2, 3, 0], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 3, 1], [2, 0, 1, 3], [1, 2, 3, 0], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0065/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 66. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[1, 3, 0, 2]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([3, 0, 2, 1], [1, 3, 0, 2], [2, 1, 0, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 0, 2, 1], [1, 3, 0, 2], [2, 1, 0, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0066/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 67. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 2, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[0, 2, 3, 1]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([0, 2, 3, 1], [0, 2, 3, 1], [0, 3, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3, 1], [0, 2, 3, 1], [0, 3, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0067/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 68. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[3, 2, 1, 0]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([1, 2, 3, 0], [3, 2, 1, 0], [2, 1, 0, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3, 0], [3, 2, 1, 0], [2, 1, 0, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0068/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 69. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 1, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[2, 1, 0, 3]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([2, 1, 3, 0], [2, 1, 0, 3], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 3, 0], [2, 1, 0, 3], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0069/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 70. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[1, 0, 3, 2]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([3, 1, 2, 0], [1, 0, 3, 2], [2, 0, 3, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1, 2, 0], [1, 0, 3, 2], [2, 0, 3, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0070/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 71. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 3, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[0, 3, 2, 1]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([0, 3, 2, 1], [0, 3, 2, 1], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 2, 1], [0, 3, 2, 1], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0071/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 72. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[3, 0, 1, 2]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([1, 3, 2, 0], [3, 0, 1, 2], [0, 2, 1, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 2, 0], [3, 0, 1, 2], [0, 2, 1, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0072/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 73. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 3, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[2, 3, 0, 1]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([2, 3, 1, 0], [2, 3, 0, 1], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 1, 0], [2, 3, 0, 1], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0073/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 74. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[1, 2, 3, 0]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([3, 2, 1, 0], [1, 2, 3, 0], [0, 3, 2, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2, 1, 0], [1, 2, 3, 0], [0, 3, 2, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0074/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 75. Two Typed Maps and Independently Audited Composition

Use complete finite topologies X=Fin(3)={0,…,2}, opens_X=[[], [0], [0, 1], [0, 1, 2]] and Y=Fin(3)={0,…,2}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]. The complete total map has images=[0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(3)={0,…,2}, opens_Z=[[], [0, 1, 2]], and g:Y→Z with images_g=[0, 1, 2]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.

View answer and explanation([0, 1, 2], [0, 1, 2], [0, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2], [0, 1, 2], Yes]])

Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2], [0, 1, 2], [0, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2], [0, 1, 2], Yes]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0075/)

Continuous maps between topological spaces · qf-ext19-continuity-composition · candidate





## 76. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 1, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([0, 1, 2, 3], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 1], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 1], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0076/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 77. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([1, 0, 2, 3], No, No, [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 1], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 3], No, No, [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 1], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0077/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 78. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 0, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([2, 0, 1, 3], No, Yes, [[[], [], Yes], [[0], [1], No], [[1], [2], Yes], [[0, 1], [1, 2], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [1, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [1], Yes], [[0, 2], [1, 2], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 3], No, Yes, [[[], [], Yes], [[0], [1], No], [[1], [2], Yes], [[0, 1], [1, 2], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [1, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [1], Yes], [[0, 2], [1, 2], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0078/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 79. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([3, 0, 1, 2], No, No, [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [2], No], [[0, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 0, 1, 2], No, No, [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [2], No], [[0, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0079/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 80. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 2, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([0, 2, 1, 3], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], Yes], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 3], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], Yes], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0080/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 81. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([1, 2, 0, 3], No, No, [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [2], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 3], No, No, [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [2], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0081/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 82. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 1, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([2, 1, 0, 3], No, Yes, [[[], [], Yes], [[0], [2], Yes], [[1], [1], No], [[0, 1], [1, 2], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [2, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [0], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 3], No, Yes, [[[], [], Yes], [[0], [2], Yes], [[1], [1], No], [[0, 1], [1, 2], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [2, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [0], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0082/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 83. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([3, 1, 0, 2], No, No, [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [2], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1, 0, 2], No, No, [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [2], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0083/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 84. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 3, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([0, 3, 1, 2], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], No], [[2], [3], No], [[0, 2], [0, 3], Yes], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 3], [0, 2], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 1, 2], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], No], [[2], [3], No], [[0, 2], [0, 3], Yes], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 3], [0, 2], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0084/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 85. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([1, 3, 0, 2], No, No, [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 0, 2], No, No, [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0085/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 86. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 3, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([2, 3, 0, 1], No, Yes, [[[], [], Yes], [[0], [2], Yes], [[1], [3], No], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [1, 2], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [0], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 0, 1], No, Yes, [[[], [], Yes], [[0], [2], Yes], [[1], [3], No], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [1, 2], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [0], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0086/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 87. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([3, 2, 0, 1], No, No, [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [1], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2, 0, 1], No, No, [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [1], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0087/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 88. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 1, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([0, 1, 3, 2], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [3], No], [[0, 2], [0, 3], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], Yes], [[3], [2], No], [[0, 3], [0, 2], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 1], [0, 1], Yes], [[0, 1, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3, 2], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [3], No], [[0, 2], [0, 3], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], Yes], [[3], [2], No], [[0, 3], [0, 2], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 1], [0, 1], Yes], [[0, 1, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0088/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 89. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([1, 0, 3, 2], No, No, [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 1], No], [[0, 1, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 3, 2], No, No, [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 1], No], [[0, 1, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0089/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 90. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 0, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([2, 0, 3, 1], No, Yes, [[[], [], Yes], [[0], [1], No], [[1], [3], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [1, 2], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 2], Yes], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [3], Yes], [[0, 2], [2, 3], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 3, 1], No, Yes, [[[], [], Yes], [[0], [1], No], [[1], [3], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [1, 2], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 2], Yes], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [3], Yes], [[0, 2], [2, 3], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0090/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 91. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([3, 0, 2, 1], No, No, [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [1], No], [[0, 3], [1, 3], No], [[0, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 0, 2, 1], No, No, [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [1], No], [[0, 3], [1, 3], No], [[0, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0091/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 92. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 2, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([0, 2, 3, 1], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], No], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], No], [[0, 3], [0, 2], Yes], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 2], [0, 3], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3, 1], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], No], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], No], [[0, 3], [0, 2], Yes], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 2], [0, 3], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0092/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 93. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([1, 2, 3, 0], No, No, [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 3], No], [[0, 1, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [2], No], [[1, 2], [2, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3, 0], No, No, [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 3], No], [[0, 1, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [2], No], [[1, 2], [2, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0093/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 94. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 1, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([2, 1, 3, 0], No, Yes, [[[], [], Yes], [[0], [3], No], [[1], [1], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [2, 3], No], [[1, 3], [1, 2], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 2], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [3], Yes], [[1, 2], [1, 3], Yes], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 3, 0], No, Yes, [[[], [], Yes], [[0], [3], No], [[1], [1], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [2, 3], No], [[1, 3], [1, 2], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 2], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [3], Yes], [[1, 2], [1, 3], Yes], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0094/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 95. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([3, 1, 2, 0], No, No, [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 1], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1, 2, 0], No, No, [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 1], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0095/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 96. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[0, 3, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([0, 3, 2, 1], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 2, 1], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0096/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 97. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([1, 3, 2, 0], No, No, [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [3], No], [[1, 3], [0, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 2, 0], No, No, [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [3], No], [[1, 3], [0, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0097/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 98. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[2, 3, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([2, 3, 1, 0], No, Yes, [[[], [], Yes], [[0], [3], No], [[1], [2], Yes], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [1], Yes], [[2, 3], [0, 1], Yes], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 1, 0], No, Yes, [[[], [], Yes], [[0], [3], No], [[1], [2], Yes], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [1], Yes], [[2, 3], [0, 1], Yes], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0098/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 99. Open Images versus Open Preimages

Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([3, 2, 1, 0], No, No, [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 1], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2, 1, 0], No, No, [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 1], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0099/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate





## 100. Open Images versus Open Preimages

Use complete finite topologies X=Fin(3)={0,…,2}, opens_X=[[], [0], [0, 1], [0, 1, 2]] and Y=Fin(3)={0,…,2}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]. The complete total map has images=[0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).

View answer and explanation([0, 1, 2], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 1], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes]], properties-differ)

Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 1], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes]], properties-differ).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261019-r03-topological-continuity-0100/)

Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261019-r03-topological-continuity-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261019-r03-topological-continuity-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261019-r03-topological-continuity-foundations/validation-report.json)