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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Continuous maps between topological spaces · qf-ext19-continuity-open-map-contrast · candidate
Canonical question JSONL · English JSONL · Validation report