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Surjective Fibers, Saturated Opens and Final-Topology Scope
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-20261015-advance-20261117-r05-quotient-topology-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1]], target carrier=[a, b] and aligned total q=[a, a, b, b, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 1]], [b, [2, 3, 4, 5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 1]], [b, [2, 3, 4, 5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
2. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2]], target carrier=[a, b] and aligned total q=[a, b, a, b, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 2]], [b, [1, 3, 4, 5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 2]], [b, [1, 3, 4, 5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
3. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2]], target carrier=[a, b] and aligned total q=[a, a, a, b, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 1, 2]], [b, [3, 4, 5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 1, 2]], [b, [3, 4, 5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
4. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3]], target carrier=[a, b] and aligned total q=[a, b, b, a, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 3]], [b, [1, 2, 4, 5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 3]], [b, [1, 2, 4, 5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
5. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3]], target carrier=[a, b] and aligned total q=[a, a, b, a, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 1, 3]], [b, [2, 4, 5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 1, 3]], [b, [2, 4, 5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
6. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3]], target carrier=[a, b] and aligned total q=[a, b, a, a, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 2, 3]], [b, [1, 4, 5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 2, 3]], [b, [1, 4, 5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
7. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3]], target carrier=[a, b] and aligned total q=[a, a, a, a, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 1, 2, 3]], [b, [4, 5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 1, 2, 3]], [b, [4, 5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
8. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4]], target carrier=[a, b] and aligned total q=[a, b, b, b, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 4]], [b, [1, 2, 3, 5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 4]], [b, [1, 2, 3, 5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
9. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4]], target carrier=[a, b] and aligned total q=[a, a, b, b, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 1, 4]], [b, [2, 3, 5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 1, 4]], [b, [2, 3, 5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
10. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 4]], target carrier=[a, b] and aligned total q=[a, b, a, b, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 2, 4]], [b, [1, 3, 5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 2, 4]], [b, [1, 3, 5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
11. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 4]], target carrier=[a, b] and aligned total q=[a, a, a, b, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 1, 2, 4]], [b, [3, 5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 1, 2, 4]], [b, [3, 5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
12. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 4]], target carrier=[a, b] and aligned total q=[a, b, b, a, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 3, 4]], [b, [1, 2, 5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 3, 4]], [b, [1, 2, 5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
13. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 4]], target carrier=[a, b] and aligned total q=[a, a, b, a, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 1, 3, 4]], [b, [2, 5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 1, 3, 4]], [b, [2, 5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
14. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 4]], target carrier=[a, b] and aligned total q=[a, b, a, a, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 2, 3, 4]], [b, [1, 5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 2, 3, 4]], [b, [1, 5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
15. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4]], target carrier=[a, b] and aligned total q=[a, a, a, a, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 1, 2, 3, 4]], [b, [5]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 1, 2, 3, 4]], [b, [5]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
16. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 5]], target carrier=[a, b] and aligned total q=[a, b, b, b, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 5]], [b, [1, 2, 3, 4]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 5]], [b, [1, 2, 3, 4]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
17. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], target carrier=[a, b] and aligned total q=[a, a, b, b, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 1, 5]], [b, [2, 3, 4]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 1, 5]], [b, [2, 3, 4]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
18. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 5]], target carrier=[a, b] and aligned total q=[a, b, a, b, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 2, 5]], [b, [1, 3, 4]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 2, 5]], [b, [1, 3, 4]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
19. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5]], target carrier=[a, b] and aligned total q=[a, a, a, b, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 1, 2, 5]], [b, [3, 4]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 1, 2, 5]], [b, [3, 4]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
20. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 5]], target carrier=[a, b] and aligned total q=[a, b, b, a, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 3, 5]], [b, [1, 2, 4]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 3, 5]], [b, [1, 2, 4]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
21. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5]], target carrier=[a, b] and aligned total q=[a, a, b, a, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 1, 3, 5]], [b, [2, 4]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 1, 3, 5]], [b, [2, 4]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
22. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 5]], target carrier=[a, b] and aligned total q=[a, b, a, a, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 2, 3, 5]], [b, [1, 4]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 2, 3, 5]], [b, [1, 4]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
23. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 5]], target carrier=[a, b] and aligned total q=[a, a, a, a, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 1, 2, 3, 5]], [b, [4]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 1, 2, 3, 5]], [b, [4]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
24. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4, 5]], target carrier=[a, b] and aligned total q=[a, b, b, b, a, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 4, 5]], [b, [1, 2, 3]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 4, 5]], [b, [1, 2, 3]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
25. A Total Surjection Has Complete Nonempty Fibers Partitioning the Source
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4, 5]], target carrier=[a, b] and aligned total q=[a, a, b, b, a, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, surjective, fiber_cover, certificate).
View answer and explanation
([[a, [0, 1, 4, 5]], [b, [2, 3]]], Yes, Yes, fibers_partition_X)
Retain every target fiber, surjectivity and complete source coverage. A missing target fiber blocks the quotient-map convention here. Do not confuse the full fiber with one representative or split a fiber when taking a full preimage. The exact result, retaining the requested order, is ([[a, [0, 1, 4, 5]], [b, [2, 3]]], Yes, Yes, fibers_partition_X).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-fibers · candidate
26. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1]], target carrier=[a, b] and aligned total q=[a, a, b, b, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 1]], [b, [2, 3, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 1]], [b, [2, 3, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
27. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2]], target carrier=[a, b] and aligned total q=[a, b, a, b, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 2]], [b, [1, 3, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 2]], [b, [1, 3, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
28. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2]], target carrier=[a, b] and aligned total q=[a, a, a, b, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 1, 2]], [b, [3, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 1, 2]], [b, [3, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
29. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3]], target carrier=[a, b] and aligned total q=[a, b, b, a, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 3]], [b, [1, 2, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 3]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 3]], [b, [1, 2, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 3]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
30. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3]], target carrier=[a, b] and aligned total q=[a, a, b, a, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 1, 3]], [b, [2, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 1, 3]], [b, [2, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
31. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3]], target carrier=[a, b] and aligned total q=[a, b, a, a, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 2, 3]], [b, [1, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 3]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 2, 3]], [b, [1, 4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 3]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
32. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3]], target carrier=[a, b] and aligned total q=[a, a, a, a, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 1, 2, 3]], [b, [4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 1, 2, 3]], [b, [4, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
33. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4]], target carrier=[a, b] and aligned total q=[a, b, b, b, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 4]], [b, [1, 2, 3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 4]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 4]], [b, [1, 2, 3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 4]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
34. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4]], target carrier=[a, b] and aligned total q=[a, a, b, b, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 1, 4]], [b, [2, 3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 4]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 1, 4]], [b, [2, 3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 4]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
35. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 4]], target carrier=[a, b] and aligned total q=[a, b, a, b, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 2, 4]], [b, [1, 3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 4]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 2, 4]], [b, [1, 3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 4]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
36. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 4]], target carrier=[a, b] and aligned total q=[a, a, a, b, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 1, 2, 4]], [b, [3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 4]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 1, 2, 4]], [b, [3, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 4]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
37. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 4]], target carrier=[a, b] and aligned total q=[a, b, b, a, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 3, 4]], [b, [1, 2, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 3, 4]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 3, 4]], [b, [1, 2, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 3, 4]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
38. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 4]], target carrier=[a, b] and aligned total q=[a, a, b, a, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 1, 3, 4]], [b, [2, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 4]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 1, 3, 4]], [b, [2, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 4]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
39. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 4]], target carrier=[a, b] and aligned total q=[a, b, a, a, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 2, 3, 4]], [b, [1, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 4]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 2, 3, 4]], [b, [1, 5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 4]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
40. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4]], target carrier=[a, b] and aligned total q=[a, a, a, a, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 1, 2, 3, 4]], [b, [5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 1, 2, 3, 4]], [b, [5]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
41. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 5]], target carrier=[a, b] and aligned total q=[a, b, b, b, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 5]], [b, [1, 2, 3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 5]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 5]], [b, [1, 2, 3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 5]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
42. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], target carrier=[a, b] and aligned total q=[a, a, b, b, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 1, 5]], [b, [2, 3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 1, 5]], [b, [2, 3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
43. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 5]], target carrier=[a, b] and aligned total q=[a, b, a, b, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 2, 5]], [b, [1, 3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 5]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 2, 5]], [b, [1, 3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 5]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
44. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5]], target carrier=[a, b] and aligned total q=[a, a, a, b, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 1, 2, 5]], [b, [3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 1, 2, 5]], [b, [3, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
45. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 5]], target carrier=[a, b] and aligned total q=[a, b, b, a, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 3, 5]], [b, [1, 2, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 3, 5]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 3, 5]], [b, [1, 2, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 3, 5]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
46. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5]], target carrier=[a, b] and aligned total q=[a, a, b, a, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 1, 3, 5]], [b, [2, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 1, 3, 5]], [b, [2, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
47. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 5]], target carrier=[a, b] and aligned total q=[a, b, a, a, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 2, 3, 5]], [b, [1, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 5]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 2, 3, 5]], [b, [1, 4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 5]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
48. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 5]], target carrier=[a, b] and aligned total q=[a, a, a, a, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 1, 2, 3, 5]], [b, [4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 5]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 1, 2, 3, 5]], [b, [4]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 5]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
49. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4, 5]], target carrier=[a, b] and aligned total q=[a, b, b, b, a, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 4, 5]], [b, [1, 2, 3]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 4, 5]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 4, 5]], [b, [1, 2, 3]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 4, 5]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
50. The Full Preimage Iff Condition Defines the Final Topology
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4, 5]], target carrier=[a, b] and aligned total q=[a, a, b, b, a, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Give (fibers, final_topology, saturated_source_opens, open_count_correspondence, quotient_certificate, certificate).
View answer and explanation
([[a, [0, 1, 4, 5]], [b, [2, 3]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 4, 5]], Yes, Yes, U_open_iff_q_preimage_U_open)
Enumerate every target subset whose full preimage is source-open, certify the resulting topology and retain the saturated source opens. Their correspondence uses saturation, not the image of an arbitrary source open. The surjection qualifies this final topology as the quotient topology in this node. The exact result, retaining the requested order, is ([[a, [0, 1, 4, 5]], [b, [2, 3]]], [[a, b], [a], []], [[], [0, 1, 2, 3, 4, 5], [0, 1, 4, 5]], Yes, Yes, U_open_iff_q_preimage_U_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-generation · candidate
51. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1]], target carrier=[a, b] and aligned total q=[a, a, b, b, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 1], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
52. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2]], target carrier=[a, b] and aligned total q=[a, b, a, b, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 1], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 1], [4, 1], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
53. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2]], target carrier=[a, b] and aligned total q=[a, a, a, b, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 1], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 1], [4, 1], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
54. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3]], target carrier=[a, b] and aligned total q=[a, b, b, a, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
55. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3]], target carrier=[a, b] and aligned total q=[a, a, b, a, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
56. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3]], target carrier=[a, b] and aligned total q=[a, b, a, a, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
57. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3]], target carrier=[a, b] and aligned total q=[a, a, a, a, b, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 0], [4, 1], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
58. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4]], target carrier=[a, b] and aligned total q=[a, b, b, b, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
59. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4]], target carrier=[a, b] and aligned total q=[a, a, b, b, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
60. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 4]], target carrier=[a, b] and aligned total q=[a, b, a, b, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
61. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 4]], target carrier=[a, b] and aligned total q=[a, a, a, b, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 1], [4, 0], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
62. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 4]], target carrier=[a, b] and aligned total q=[a, b, b, a, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
63. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 4]], target carrier=[a, b] and aligned total q=[a, a, b, a, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
64. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 4]], target carrier=[a, b] and aligned total q=[a, b, a, a, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
65. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4]], target carrier=[a, b] and aligned total q=[a, a, a, a, a, b]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 0], [4, 0], [5, 1]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
66. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 5]], target carrier=[a, b] and aligned total q=[a, b, b, b, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
67. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 5]], target carrier=[a, b] and aligned total q=[a, a, b, b, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
68. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 5]], target carrier=[a, b] and aligned total q=[a, b, a, b, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
69. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5]], target carrier=[a, b] and aligned total q=[a, a, a, b, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 1], [4, 1], [5, 0]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
70. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 5]], target carrier=[a, b] and aligned total q=[a, b, b, a, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
71. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5]], target carrier=[a, b] and aligned total q=[a, a, b, a, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
72. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 5]], target carrier=[a, b] and aligned total q=[a, b, a, a, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 0], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
73. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 5]], target carrier=[a, b] and aligned total q=[a, a, a, a, b, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 0], [3, 0], [4, 1], [5, 0]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
74. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4, 5]], target carrier=[a, b] and aligned total q=[a, b, b, b, a, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 1], [4, 0], [5, 0]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 1], [2, 1], [3, 1], [4, 0], [5, 0]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
75. The Final-Topology Criterion Compares h with Its Composite with q
Use complete source points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4, 5]], target carrier=[a, b] and aligned total q=[a, a, b, b, a, a]. Normal quotient families require q surjective. Target opens are exactly the subsets with open full preimage; continuity alone is weaker. Target Z=[0, 1], topology=[[], [1], [0, 1]], aligned h=[0, 1]. Give (h_continuous, composite_continuous, criterion_equivalence, h_rows, composite_rows, certificate).
h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology.
View answer and explanation
(No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 0], [5, 0]], quotient_final_property_for_surjective_q)
Audit every target-open preimage for h and h∘q with their complete aligned tables. Both maps can be discontinuous while the equivalence holds. The qualified surjection and full final topology precede this criterion. The exact result, retaining the requested order, is (No, No, Yes, [[a, 0], [b, 1]], [[0, 0], [1, 0], [2, 1], [3, 1], [4, 0], [5, 0]], quotient_final_property_for_surjective_q).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-final-continuity · candidate
76. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=non_surjective and complete model={"x_points":[0,1,2,3],"x_topology":[[],[0],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]}. Give (mode, fibers, surjective, final_topology, final_topology_exists, quotient_map_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y4], [y0], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y4], [y0], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
77. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=open_image_error and complete model={"x_points":[0,1,2,3,4],"x_topology":[[],[0],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","b","b"],"subset":[0]}. Give (mode, source_subset, source_open, image, image_preimage, image_open, automatic_open_map_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(open_image_error, [0], Yes, [a], [0, 1, 2], No, No, quotient_maps_need_not_be_open)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (open_image_error, [0], Yes, [a], [0, 1, 2], No, No, quotient_maps_need_not_be_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
78. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=coarser_target and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2]],"y_points":["a","b"],"q_map":["a","a","a","b","b","b"],"candidate_topology":[[],["a","b"]]}. Give (mode, candidate_topology, final_topology, q_continuous, properly_coarser, final_topology_identification_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(coarser_target, [[a, b], []], [[a, b], [a], []], Yes, Yes, No, continuity_alone_does_not_identify_the_final_topology)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (coarser_target, [[a, b], []], [[a, b], [a], []], Yes, Yes, No, continuity_alone_does_not_identify_the_final_topology).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
79. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=fiber_split and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5]],"y_points":["a","b"],"q_map":["a","a","b","b","b","b"],"subset":[0]}. Give (mode, source_subset, image, saturation, saturated, full_preimage_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(fiber_split, [0], [a], [0, 1], No, No, only_saturated_subsets_are_full_preimages)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (fiber_split, [0], [a], [0, 1], No, No, only_saturated_subsets_are_full_preimages).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
80. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=non_surjective and complete model={"x_points":[0,1,2,3],"x_topology":[[],[1],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]}. Give (mode, fibers, surjective, final_topology, final_topology_exists, quotient_map_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y1, y4], [y1], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y1, y4], [y1], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
81. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=open_image_error and complete model={"x_points":[0,1,2,3,4],"x_topology":[[],[1],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","b","b"],"subset":[1]}. Give (mode, source_subset, source_open, image, image_preimage, image_open, automatic_open_map_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(open_image_error, [1], Yes, [a], [0, 1, 2], No, No, quotient_maps_need_not_be_open)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (open_image_error, [1], Yes, [a], [0, 1, 2], No, No, quotient_maps_need_not_be_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
82. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=coarser_target and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3]],"y_points":["a","b"],"q_map":["a","a","a","a","b","b"],"candidate_topology":[[],["a","b"]]}. Give (mode, candidate_topology, final_topology, q_continuous, properly_coarser, final_topology_identification_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(coarser_target, [[a, b], []], [[a, b], [a], []], Yes, Yes, No, continuity_alone_does_not_identify_the_final_topology)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (coarser_target, [[a, b], []], [[a, b], [a], []], Yes, Yes, No, continuity_alone_does_not_identify_the_final_topology).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
83. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=fiber_split and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5]],"y_points":["a","b"],"q_map":["a","b","a","b","b","b"],"subset":[0]}. Give (mode, source_subset, image, saturation, saturated, full_preimage_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(fiber_split, [0], [a], [0, 2], No, No, only_saturated_subsets_are_full_preimages)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (fiber_split, [0], [a], [0, 2], No, No, only_saturated_subsets_are_full_preimages).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
84. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=non_surjective and complete model={"x_points":[0,1,2,3],"x_topology":[[],[0,1],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]}. Give (mode, fibers, surjective, final_topology, final_topology_exists, quotient_map_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y1, y4], [y0, y1], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y1, y4], [y0, y1], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
85. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=open_image_error and complete model={"x_points":[0,1,2,3,4],"x_topology":[[],[0,1],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","b","b"],"subset":[0,1]}. Give (mode, source_subset, source_open, image, image_preimage, image_open, automatic_open_map_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(open_image_error, [0, 1], Yes, [a], [0, 1, 2], No, No, quotient_maps_need_not_be_open)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (open_image_error, [0, 1], Yes, [a], [0, 1, 2], No, No, quotient_maps_need_not_be_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
86. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=coarser_target and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,4]],"y_points":["a","b"],"q_map":["a","a","a","b","a","b"],"candidate_topology":[[],["a","b"]]}. Give (mode, candidate_topology, final_topology, q_continuous, properly_coarser, final_topology_identification_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(coarser_target, [[a, b], []], [[a, b], [a], []], Yes, Yes, No, continuity_alone_does_not_identify_the_final_topology)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (coarser_target, [[a, b], []], [[a, b], [a], []], Yes, Yes, No, continuity_alone_does_not_identify_the_final_topology).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
87. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=fiber_split and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5]],"y_points":["a","b"],"q_map":["a","a","a","b","b","b"],"subset":[0]}. Give (mode, source_subset, image, saturation, saturated, full_preimage_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(fiber_split, [0], [a], [0, 1, 2], No, No, only_saturated_subsets_are_full_preimages)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (fiber_split, [0], [a], [0, 1, 2], No, No, only_saturated_subsets_are_full_preimages).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
88. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=non_surjective and complete model={"x_points":[0,1,2,3],"x_topology":[[],[2],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]}. Give (mode, fibers, surjective, final_topology, final_topology_exists, quotient_map_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y2, y4], [y2], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y2, y4], [y2], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
89. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=open_image_error and complete model={"x_points":[0,1,2,3,4],"x_topology":[[],[2],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","b","b"],"subset":[2]}. Give (mode, source_subset, source_open, image, image_preimage, image_open, automatic_open_map_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(open_image_error, [2], Yes, [a], [0, 1, 2], No, No, quotient_maps_need_not_be_open)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (open_image_error, [2], Yes, [a], [0, 1, 2], No, No, quotient_maps_need_not_be_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
90. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=coarser_target and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","a","a","b"],"candidate_topology":[[],["a","b"]]}. Give (mode, candidate_topology, final_topology, q_continuous, properly_coarser, final_topology_identification_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(coarser_target, [[a, b], []], [[a, b], [a], []], Yes, Yes, No, continuity_alone_does_not_identify_the_final_topology)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (coarser_target, [[a, b], []], [[a, b], [a], []], Yes, Yes, No, continuity_alone_does_not_identify_the_final_topology).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
91. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=fiber_split and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5]],"y_points":["a","b"],"q_map":["a","b","b","a","b","b"],"subset":[0]}. Give (mode, source_subset, image, saturation, saturated, full_preimage_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(fiber_split, [0], [a], [0, 3], No, No, only_saturated_subsets_are_full_preimages)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (fiber_split, [0], [a], [0, 3], No, No, only_saturated_subsets_are_full_preimages).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
92. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=non_surjective and complete model={"x_points":[0,1,2,3],"x_topology":[[],[0,2],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]}. Give (mode, fibers, surjective, final_topology, final_topology_exists, quotient_map_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y2, y4], [y0, y2], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y2, y4], [y0, y2], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
93. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=open_image_error and complete model={"x_points":[0,1,2,3,4],"x_topology":[[],[0,2],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","b","b"],"subset":[0,2]}. Give (mode, source_subset, source_open, image, image_preimage, image_open, automatic_open_map_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(open_image_error, [0, 2], Yes, [a], [0, 1, 2], No, No, quotient_maps_need_not_be_open)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (open_image_error, [0, 2], Yes, [a], [0, 1, 2], No, No, quotient_maps_need_not_be_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
94. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=coarser_target and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,5]],"y_points":["a","b"],"q_map":["a","a","a","b","b","a"],"candidate_topology":[[],["a","b"]]}. Give (mode, candidate_topology, final_topology, q_continuous, properly_coarser, final_topology_identification_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(coarser_target, [[a, b], []], [[a, b], [a], []], Yes, Yes, No, continuity_alone_does_not_identify_the_final_topology)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (coarser_target, [[a, b], []], [[a, b], [a], []], Yes, Yes, No, continuity_alone_does_not_identify_the_final_topology).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
95. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=fiber_split and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5]],"y_points":["a","b"],"q_map":["a","a","b","a","b","b"],"subset":[0]}. Give (mode, source_subset, image, saturation, saturated, full_preimage_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(fiber_split, [0], [a], [0, 1, 3], No, No, only_saturated_subsets_are_full_preimages)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (fiber_split, [0], [a], [0, 1, 3], No, No, only_saturated_subsets_are_full_preimages).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
96. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=non_surjective and complete model={"x_points":[0,1,2,3],"x_topology":[[],[1,2],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]}. Give (mode, fibers, surjective, final_topology, final_topology_exists, quotient_map_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y1, y2, y4], [y1, y2], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y1, y2, y4], [y1, y2], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
97. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=open_image_error and complete model={"x_points":[0,1,2,3,4],"x_topology":[[],[1,2],[0,1,2,3,4]],"y_points":["a","b"],"q_map":["a","a","a","b","b"],"subset":[1,2]}. Give (mode, source_subset, source_open, image, image_preimage, image_open, automatic_open_map_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(open_image_error, [1, 2], Yes, [a], [0, 1, 2], No, No, quotient_maps_need_not_be_open)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (open_image_error, [1, 2], Yes, [a], [0, 1, 2], No, No, quotient_maps_need_not_be_open).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
98. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=coarser_target and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5],[0,1,2,3,5]],"y_points":["a","b"],"q_map":["a","a","a","a","b","a"],"candidate_topology":[[],["a","b"]]}. Give (mode, candidate_topology, final_topology, q_continuous, properly_coarser, final_topology_identification_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(coarser_target, [[a, b], []], [[a, b], [a], []], Yes, Yes, No, continuity_alone_does_not_identify_the_final_topology)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (coarser_target, [[a, b], []], [[a, b], [a], []], Yes, Yes, No, continuity_alone_does_not_identify_the_final_topology).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
99. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=fiber_split and complete model={"x_points":[0,1,2,3,4,5],"x_topology":[[],[0,1,2,3,4,5]],"y_points":["a","b"],"q_map":["a","b","a","a","b","b"],"subset":[0]}. Give (mode, source_subset, image, saturation, saturated, full_preimage_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(fiber_split, [0], [a], [0, 2, 3], No, No, only_saturated_subsets_are_full_preimages)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (fiber_split, [0], [a], [0, 2, 3], No, No, only_saturated_subsets_are_full_preimages).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
100. Final Topology Can Exist without Surjectivity, while Quotient-Map Qualification Still Fails
Use mode=non_surjective and complete model={"x_points":[0,1,2,3],"x_topology":[[],[0,1,2],[0,1,2,3]],"y_points":["y0","y1","y2","y3","y4"],"q_map":["y0","y1","y2","y3"]}. Give (mode, fibers, surjective, final_topology, final_topology_exists, quotient_map_claim, boundary).
The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology.
View answer and explanation
(non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y1, y2, y4], [y0, y1, y2], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention)
A nonsurjection still defines a final topology by the full preimage iff condition, but it is not a quotient map under this surjective convention. Open images need not be open; a properly coarser target can keep q continuous without being final. A fiber-splitting subset is not a full preimage. The preimage-if-and-only-if definition uniquely determines the final topology even for a nonsurjection; continuity alone does not. Retain the separate surjective quotient-map naming convention. The exact result, retaining the requested order, is (non_surjective, [[y0, [0]], [y1, [1]], [y2, [2]], [y3, [3]], [y4, []]], No, [[y0, y1, y2, y3, y4], [y0, y1, y2, y3], [y0, y1, y2, y4], [y0, y1, y2], [y4], []], Yes, No, preimage_defined_final_topology_exists_but_map_is_not_a_quotient_map_by_this_convention).
Reviewed step-by-step derivation
Quotient topology · qf-mod48-quotient-topology-boundary · candidate
Canonical question JSONL · English JSONL · Validation report