# Question 21206: Relative-Open but Not Ambient-Open Is a Valid Boundary Case

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0041-21206 · q-20261014-advance-20261108-r03-subspace-topology-0081



## Original question and givens

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).


- G001 — 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).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-subspace-topology](https://ocme.evemisslab.com/en/mko/mko-subspace-topology/)





## Step-by-step derivation

S001 · impliesBefore

source operation=subtop-ambient-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[candidate, relative_open, ambient_open, intersection_certificate, boundary]

Conditions: none

Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ORIGINAL parent Fin(5);parent open index=0,U=[];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]]

After

ALL [other_open_index,union,intersection] rows=[[0, [], []], [1, [0], []], [2, [0, 1], []], [3, [0, 1, 2, 3, 4], []], [4, [0, 1, 3, 4], []], [5, [0, 1, 4], []]];each belongs to the SAME parent family

Conditions: none

First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.

Step identifiers and dependenciesRule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S001





S003 · impliesBefore

ORIGINAL parent Fin(5);parent open index=1,U=[0];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]]

After

ALL [other_open_index,union,intersection] rows=[[0, [0], []], [1, [0], [0]], [2, [0, 1], [0]], [3, [0, 1, 2, 3, 4], [0]], [4, [0, 1, 3, 4], [0]], [5, [0, 1, 4], [0]]];each belongs to the SAME parent family

Conditions: none

First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.

Step identifiers and dependenciesRule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S002





S004 · impliesBefore

ORIGINAL parent Fin(5);parent open index=2,U=[0, 1];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]]

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 1], []], [1, [0, 1], [0]], [2, [0, 1], [0, 1]], [3, [0, 1, 2, 3, 4], [0, 1]], [4, [0, 1, 3, 4], [0, 1]], [5, [0, 1, 4], [0, 1]]];each belongs to the SAME parent family

Conditions: none

First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.

Step identifiers and dependenciesRule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S003





S005 · impliesBefore

ORIGINAL parent Fin(5);parent open index=3,U=[0, 1, 2, 3, 4];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]]

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2, 3, 4], []], [1, [0, 1, 2, 3, 4], [0]], [2, [0, 1, 2, 3, 4], [0, 1]], [3, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4]], [4, [0, 1, 2, 3, 4], [0, 1, 3, 4]], [5, [0, 1, 2, 3, 4], [0, 1, 4]]];each belongs to the SAME parent family

Conditions: none

First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.

Step identifiers and dependenciesRule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S004





S006 · impliesBefore

ORIGINAL parent Fin(5);parent open index=4,U=[0, 1, 3, 4];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]]

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 3, 4], []], [1, [0, 1, 3, 4], [0]], [2, [0, 1, 3, 4], [0, 1]], [3, [0, 1, 2, 3, 4], [0, 1, 3, 4]], [4, [0, 1, 3, 4], [0, 1, 3, 4]], [5, [0, 1, 3, 4], [0, 1, 4]]];each belongs to the SAME parent family

Conditions: none

First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.

Step identifiers and dependenciesRule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S005





S007 · impliesBefore

ORIGINAL parent Fin(5);parent open index=5,U=[0, 1, 4];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]]

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 4], []], [1, [0, 1, 4], [0]], [2, [0, 1, 4], [0, 1]], [3, [0, 1, 2, 3, 4], [0, 1, 4]], [4, [0, 1, 3, 4], [0, 1, 4]], [5, [0, 1, 4], [0, 1, 4]]];each belongs to the SAME parent family

Conditions: none

First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.

Step identifiers and dependenciesRule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S006





S008 · impliesBefore

ORIGINAL parent open index=0,U=[];ORIGINAL subset Y=[1, 2, 3]

After

actual U intersect Y=[];retain parent index even if another open has the same intersection

Conditions: none

Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.

Step identifiers and dependenciesRule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S007





S009 · impliesBefore

ORIGINAL parent open index=1,U=[0];ORIGINAL subset Y=[1, 2, 3]

After

actual U intersect Y=[];retain parent index even if another open has the same intersection

Conditions: none

Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.

Step identifiers and dependenciesRule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S008





S010 · impliesBefore

ORIGINAL parent open index=2,U=[0, 1];ORIGINAL subset Y=[1, 2, 3]

After

actual U intersect Y=[1];retain parent index even if another open has the same intersection

Conditions: none

Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.

Step identifiers and dependenciesRule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S009





S011 · impliesBefore

ORIGINAL parent open index=3,U=[0, 1, 2, 3, 4];ORIGINAL subset Y=[1, 2, 3]

After

actual U intersect Y=[1, 2, 3];retain parent index even if another open has the same intersection

Conditions: none

Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.

Step identifiers and dependenciesRule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S010





S012 · impliesBefore

ORIGINAL parent open index=4,U=[0, 1, 3, 4];ORIGINAL subset Y=[1, 2, 3]

After

actual U intersect Y=[1, 3];retain parent index even if another open has the same intersection

Conditions: none

Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.

Step identifiers and dependenciesRule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S011





S013 · impliesBefore

ORIGINAL parent open index=5,U=[0, 1, 4];ORIGINAL subset Y=[1, 2, 3]

After

actual U intersect Y=[1];retain parent index even if another open has the same intersection

Conditions: none

Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.

Step identifiers and dependenciesRule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S012





S014 · calculationBefore

subtop distinct relative open count := sum(1, 1, 1, 1)

After

subtop distinct relative open count = 4

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · impliesBefore

ORIGINAL Y=[1, 2, 3];ALL parent intersections=[[], [], [1], [1, 2, 3], [1, 3], [1]]

After

normalized relative topology=[[], [1], [1, 2, 3], [1, 3]];distinct count=4;empty and Y included;unions and finite intersections commute with intersect Y

Conditions: none

Intersection with Y commutes with arbitrary unions and finite intersections;parent closure gives the induced axioms.For empty Y every intersection is empty,so the topology has one open,not zero.

Step identifiers and dependenciesRule: subtop_induced_axioms_and_empty_Y · Uses: G001, S014





S016 · impliesBefore

ORIGINAL candidate=[1, 2, 3] subset Y=[1, 2, 3];ALL original parent opens=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]]

After

ALL ambient U with U intersect Y=candidate=[[0, 1, 2, 3, 4]];relative_open=Yes;ambient_open=No;Y minus candidate=[];relative_closed=Yes

Conditions: none

Relative openness is witnessed by ambient-open intersections,not ambient membership of the candidate.Relative closedness tests the complement inside Y.De Morgan gives the ambient-closed-intersection form;do not substitute X minus candidate for the relative complement.

Step identifiers and dependenciesRule: subtop_relative_open_witnesses_and_relative_complement · Uses: G001, S015





S017 · impliesBefore

ACTUAL candidate=[1, 2, 3];relative_open=Yes;ambient_open=No

After

relative-only certificate=Yes;O intersect Y definition does NOT require O subset Y

Conditions: none

This boundary requires an actually relative-open but non-ambient-open candidate;it does not say every relative open fails ambient openness.

Step identifiers and dependenciesRule: subtop_boundary_requires_actual_relative_not_ambient · Uses: G001, S016









## Reverse checks


- S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S014 · premise_and_result_audit — Recompute subtop distinct relative open count with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

([1, 2, 3], Yes, No, Yes, O_intersect_Y_not_O_subset_Y)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 9978c73fe66e08a1107e819e779e73143eb8fb6c769b93d5d0e036afe6932b93
- candidate record SHA-256: 142000349f93bd787ab483205d66ef47f844713d038c1ee4b10f1f5243f656ce
- semantic review receipt SHA-256: 8f134403487694af4c70d2e00ec84bab542043ef4b3c54517c04f3c409c0262a
- question-first witness SHA-256: ad772845088ed8e99f87f90d4264b05ea97be35775305a02011af5ee35f0f4ce
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261108-r03-subspace-topology-0081.json)