Complete Relative Opens, Relative Complements and Ambient Boundaries
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batch-20261014-advance-20261108-r03-subspace-topology-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 3, 4], [[], [1], [1, 3], [1, 3, 4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 3, 4], [[], [1], [1, 3], [1, 3, 4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
2. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 3, 4], [[], [1], [1, 3, 4], [1, 4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 3, 4], [[], [1], [1, 3, 4], [1, 4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
3. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 2, 4], [[], [1], [1, 2], [1, 2, 4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 4], [[], [1], [1, 2], [1, 2, 4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
4. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 2, 4], [[], [1], [1, 2, 4], [1, 4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 4], [[], [1], [1, 2, 4], [1, 4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
5. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 2, 3], [[], [1], [1, 2], [1, 2, 3]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 3], [[], [1], [1, 2], [1, 2, 3]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
6. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 2, 3], [[], [1], [1, 2, 3], [1, 3]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 3], [[], [1], [1, 2, 3], [1, 3]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
7. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([2, 3, 4], [[], [2], [2, 3], [2, 3, 4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([2, 3, 4], [[], [2], [2, 3], [2, 3, 4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
8. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([2, 3, 4], [[], [2], [2, 3, 4], [2, 4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([2, 3, 4], [[], [2], [2, 3, 4], [2, 4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
9. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 2, 4], [[], [1, 2], [1, 2, 4], [2]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 4], [[], [1, 2], [1, 2, 4], [2]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
10. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 2, 4], [[], [1, 2, 4], [2], [2, 4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 4], [[], [1, 2, 4], [2], [2, 4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
11. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 2, 3], [[], [1, 2], [1, 2, 3], [2]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 3], [[], [1, 2], [1, 2, 3], [2]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
12. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 2, 3], [[], [1, 2, 3], [2], [2, 3]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 3], [[], [1, 2, 3], [2], [2, 3]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
13. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([2, 3, 4], [[], [2, 3], [2, 3, 4], [3]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([2, 3, 4], [[], [2, 3], [2, 3, 4], [3]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
14. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([2, 3, 4], [[], [2, 3, 4], [3], [3, 4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([2, 3, 4], [[], [2, 3, 4], [3], [3, 4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
15. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 3, 4], [[], [1, 3], [1, 3, 4], [3]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 3, 4], [[], [1, 3], [1, 3, 4], [3]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
16. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 3, 4], [[], [1, 3, 4], [3], [3, 4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 3, 4], [[], [1, 3, 4], [3], [3, 4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
17. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 2, 3], [[], [1, 2, 3], [1, 3], [3]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 3], [[], [1, 2, 3], [1, 3], [3]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
18. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 2, 3], [[], [1, 2, 3], [2, 3], [3]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 3], [[], [1, 2, 3], [2, 3], [3]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
19. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([2, 3, 4], [[], [2, 3, 4], [2, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([2, 3, 4], [[], [2, 3, 4], [2, 4], [4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
20. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([2, 3, 4], [[], [2, 3, 4], [3, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([2, 3, 4], [[], [2, 3, 4], [3, 4], [4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
21. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 3, 4], [[], [1, 3, 4], [1, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 3, 4], [[], [1, 3, 4], [1, 4], [4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
22. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 3, 4], [[], [1, 3, 4], [3, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 3, 4], [[], [1, 3, 4], [3, 4], [4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
23. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 2, 4], [[], [1, 2, 4], [1, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 4], [[], [1, 2, 4], [1, 4], [4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
24. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([1, 2, 4], [[], [1, 2, 4], [2, 4], [4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([1, 2, 4], [[], [1, 2, 4], [2, 4], [4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
25. Normalize All Ambient-Open Intersections to Obtain the Relative Topology
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], and subset Y=[0, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).
View answer and explanation
([0, 3, 4], [[], [0], [0, 3], [0, 3, 4]], 4, Yes, all_parent_open_intersections_normalized)
Intersect every parent open with Y, identify duplicate intersections and retain the source canonical family order. The empty subspace has the one-open-set topology {empty set}. This finite complete construction does not sample an unspecified ambient topology. The exact result, retaining the requested order, is ([0, 3, 4], [[], [0], [0, 3], [0, 3, 4]], 4, Yes, all_parent_open_intersections_normalized).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-induced-topology · candidate
26. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([1], Yes, [[0, 1], [0, 1, 2]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([1], Yes, [[0, 1], [0, 1, 2]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
27. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([1], Yes, [[0, 1], [0, 1, 2]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([1], Yes, [[0, 1], [0, 1, 2]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
28. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([1], Yes, [[0, 1], [0, 1, 3]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([1], Yes, [[0, 1], [0, 1, 3]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
29. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([1], Yes, [[0, 1], [0, 1, 3]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([1], Yes, [[0, 1], [0, 1, 3]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
30. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([1], Yes, [[0, 1], [0, 1, 4]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([1], Yes, [[0, 1], [0, 1, 4]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
31. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([1], Yes, [[0, 1], [0, 1, 4]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([1], Yes, [[0, 1], [0, 1, 4]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
32. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([2], Yes, [[0, 1, 2], [0, 2]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([2], Yes, [[0, 1, 2], [0, 2]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
33. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([2], Yes, [[0, 1, 2], [0, 2]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([2], Yes, [[0, 1, 2], [0, 2]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
34. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([2], Yes, [[0, 2], [0, 2, 3]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([2], Yes, [[0, 2], [0, 2, 3]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
35. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([2], Yes, [[0, 2], [0, 2, 3]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([2], Yes, [[0, 2], [0, 2, 3]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
36. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([2], Yes, [[0, 2], [0, 2, 4]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([2], Yes, [[0, 2], [0, 2, 4]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
37. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([2], Yes, [[0, 2], [0, 2, 4]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([2], Yes, [[0, 2], [0, 2, 4]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
38. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([3], Yes, [[0, 1, 3], [0, 3]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([3], Yes, [[0, 1, 3], [0, 3]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
39. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([3], Yes, [[0, 1, 3], [0, 3]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([3], Yes, [[0, 1, 3], [0, 3]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
40. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([3], Yes, [[0, 2, 3], [0, 3]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([3], Yes, [[0, 2, 3], [0, 3]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
41. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([3], Yes, [[0, 2, 3], [0, 3]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([3], Yes, [[0, 2, 3], [0, 3]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
42. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([3], Yes, [[0, 3], [0, 3, 4]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([3], Yes, [[0, 3], [0, 3, 4]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
43. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([3], Yes, [[0, 3], [0, 3, 4]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([3], Yes, [[0, 3], [0, 3, 4]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
44. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[4] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([4], Yes, [[0, 1, 4], [0, 4]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([4], Yes, [[0, 1, 4], [0, 4]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
45. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[4] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([4], Yes, [[0, 1, 4], [0, 4]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([4], Yes, [[0, 1, 4], [0, 4]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
46. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[4] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([4], Yes, [[0, 2, 4], [0, 4]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([4], Yes, [[0, 2, 4], [0, 4]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
47. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[4] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([4], Yes, [[0, 2, 4], [0, 4]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([4], Yes, [[0, 2, 4], [0, 4]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
48. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[4] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([4], Yes, [[0, 3, 4], [0, 4]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([4], Yes, [[0, 3, 4], [0, 4]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
49. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[4] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([4], Yes, [[0, 3, 4], [0, 4]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([4], Yes, [[0, 3, 4], [0, 4]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
50. A Relative Open May Need an Ambient Open Extending beyond Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], and subset Y=[0, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[0] is a duplicate-free subset of Y. Give (candidate, relative_open, parent_open_witnesses, boundary).
View answer and explanation
([0], Yes, [[0, 1], [0, 1, 2]], relative_open_need_not_be_ambient_open)
Check membership in the induced topology, then list every ambient open U with U∩Y equal to the candidate. U need not be contained in Y, and relative openness need not be ambient openness. The exact result, retaining the requested order, is ([0], Yes, [[0, 1], [0, 1, 2]], relative_open_need_not_be_ambient_open).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-open-audit · candidate
51. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([3, 4], [1], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([3, 4], [1], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
52. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([3, 4], [1], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([3, 4], [1], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
53. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([2, 4], [1], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 4], [1], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
54. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([2, 4], [1], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 4], [1], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
55. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([2, 3], [1], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 3], [1], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
56. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([2, 3], [1], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 3], [1], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
57. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([3, 4], [2], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([3, 4], [2], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
58. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([3, 4], [2], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([3, 4], [2], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
59. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([1, 4], [2], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 4], [2], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
60. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([1, 4], [2], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 4], [2], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
61. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([1, 3], [2], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 3], [2], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
62. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([1, 3], [2], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 3], [2], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
63. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([2, 4], [3], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 4], [3], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
64. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([2, 4], [3], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 4], [3], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
65. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([1, 4], [3], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 4], [3], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
66. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([1, 4], [3], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 4], [3], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
67. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([1, 2], [3], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 2], [3], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
68. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([1, 2], [3], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 2], [3], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
69. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([2, 3], [4], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 3], [4], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
70. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([2, 3], [4], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([2, 3], [4], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
71. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([1, 3], [4], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 3], [4], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
72. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([1, 3], [4], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 3], [4], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
73. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([1, 2], [4], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 2], [4], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
74. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([1, 2], [4], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([1, 2], [4], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
75. Relative Closedness Uses the Complement inside Y
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], and subset Y=[0, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[3, 4] is a duplicate-free subset of Y. Give (candidate, relative_complement, relative_closed, boundary).
relative_complement is taken inside Y, not inside the ambient carrier X.
View answer and explanation
([3, 4], [0], Yes, complement_taken_inside_Y)
Compute Y minus the candidate and check that this relative complement is open in the induced topology. Do not silently replace it by X minus the candidate or infer ambient closedness. The exact result, retaining the requested order, is ([3, 4], [0], Yes, complement_taken_inside_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-relative-closed-audit · candidate
76. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
77. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
78. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
79. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
80. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 3] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
81. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 3] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
82. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
83. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
84. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
85. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
86. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 3] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
87. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 3] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
88. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
89. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
90. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
91. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
92. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 3] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
93. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], and subset Y=[1, 2, 3]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 3] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
94. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
95. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], and subset Y=[2, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[2, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
96. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
97. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], and subset Y=[1, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
98. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
99. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[1, 2, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
100. Relative-Open but Not Ambient-Open Is a Valid Boundary Case
Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], and subset Y=[0, 3, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Candidate=[0, 3, 4] is a duplicate-free subset of Y. Give (candidate, relative_open, ambient_open, intersection_certificate, boundary).
View answer and explanation
([0, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)
Require the actual candidate to be relatively open but not ambient open. The parent-open intersection certificate does not require its ambient witness U to lie inside Y. The exact result, retaining the requested order, is ([0, 3, 4], Yes, No, Yes, O_intersect_Y_not_O_subset_Y).
Reviewed step-by-step derivation
Subspace topology · qf-mod39-subtop-ambient-boundary · candidate
Canonical question JSONL · English JSONL · Validation report