# Complete Relative Opens, Relative Complements and Ambient Boundaries

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

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

batch-20261014-advance-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](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r03-subspace-topology-0100/)

Subspace topology · qf-mod39-subtop-ambient-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261108-r03-subspace-topology-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261108-r03-subspace-topology-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261108-r03-subspace-topology-foundations/validation-report.json)