# Question 21644: Compactness Does Not Supply General Ambient Closedness or Boundedness

[Return to catalogue page 109](https://ocme.evemisslab.com/en/questions/page/109/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261109-r03-compact-subset-foundations/#q-20261014-advance-20261109-r03-compact-subset-0094)

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.

R0042-21644 · q-20261014-advance-20261109-r03-compact-subset-0094



## Original question and givens

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]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).


- G001 — 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]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, 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-compact-subset](https://ocme.evemisslab.com/en/mko/mko-compact-subset/)





## Step-by-step derivation

S001 · impliesBefore

source operation=compact-nonproperty-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary]

Conditions: none

Keep the original historical interface marker while listing ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their source 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, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]]

After

ALL [other_open_index,union,intersection] rows=[[0, [], []], [1, [0], []], [2, [0, 1, 2, 3, 4], []], [3, [0, 1, 2, 4], []], [4, [0, 1, 4], []], [5, [0, 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, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]]

After

ALL [other_open_index,union,intersection] rows=[[0, [0], []], [1, [0], [0]], [2, [0, 1, 2, 3, 4], [0]], [3, [0, 1, 2, 4], [0]], [4, [0, 1, 4], [0]], [5, [0, 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, 2, 3, 4];complete topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 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, 2, 3, 4]], [3, [0, 1, 2, 3, 4], [0, 1, 2, 4]], [4, [0, 1, 2, 3, 4], [0, 1, 4]], [5, [0, 1, 2, 3, 4], [0, 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, S003





S005 · impliesBefore

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

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2, 4], []], [1, [0, 1, 2, 4], [0]], [2, [0, 1, 2, 3, 4], [0, 1, 2, 4]], [3, [0, 1, 2, 4], [0, 1, 2, 4]], [4, [0, 1, 2, 4], [0, 1, 4]], [5, [0, 1, 2, 4], [0, 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, 4];complete topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]]

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 4], []], [1, [0, 1, 4], [0]], [2, [0, 1, 2, 3, 4], [0, 1, 4]], [3, [0, 1, 2, 4], [0, 1, 4]], [4, [0, 1, 4], [0, 1, 4]], [5, [0, 1, 4], [0, 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, 4];complete topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]]

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 4], []], [1, [0, 4], [0]], [2, [0, 1, 2, 3, 4], [0, 4]], [3, [0, 1, 2, 4], [0, 4]], [4, [0, 1, 4], [0, 4]], [5, [0, 4], [0, 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=[2, 3, 4]

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=[2, 3, 4]

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, 2, 3, 4];ORIGINAL subset Y=[2, 3, 4]

After

actual U intersect Y=[2, 3, 4];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, 4];ORIGINAL subset Y=[2, 3, 4]

After

actual U intersect Y=[2, 4];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, 4];ORIGINAL subset Y=[2, 3, 4]

After

actual U intersect Y=[4];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, 4];ORIGINAL subset Y=[2, 3, 4]

After

actual U intersect Y=[4];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=[2, 3, 4];ALL parent intersections=[[], [], [2, 3, 4], [2, 4], [4], [4]]

After

normalized relative topology=[[], [2, 3, 4], [2, 4], [4]];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 K=[2, 3, 4];COMPLETE finite relative open family=[[], [2, 3, 4], [2, 4], [4]];distinct open count=4

After

for ANY relative-open cover of K,retain one used original index per DISTINCT open;at most this many opens form a finite subcover;empty K covered by the empty subfamily

Conditions: none

This argument covers every cover,not one sample.Even infinitely repeated indices use only finitely many distinct opens;choose one original index for each used open.The source finite tag concerns distinct opens,not the number of indices in every indexed cover.

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





S017 · impliesBefore

ACTUAL finite K=[2, 3, 4] is compact in its relative topology

After

two fixed No markers WITHHOLD general ambient-closedness and boundedness claims;no metric or Hausdorff premise is supplied

Conditions: none

Finite-model compactness is not a general closed-and-bounded equivalence.The two No markers do not classify this K as nonclosed or unbounded;special cases such as empty K remain possible.

Step identifiers and dependenciesRule: compact_general_closed_bounded_nonclaims_not_instance_negations · 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

([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)

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: b107e34bd2a73f8f7fd6c9131fdeb79fd85d04d5aab5143872913468626695b3
- candidate record SHA-256: 7573c78782ebd8efe0b57b1c4010f3826d82a5e476409499aa3adc9d7561586a
- semantic review receipt SHA-256: dd44555041dfbb6fe9936aa00b9a5f8c5982c2784ef61dd0cbd5d1a303a49c1a
- question-first witness SHA-256: 9a6011263f6ac53b9a1fd57addf2f3e2f6088b54802c539f332d689480dc2d17
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261109-r03-compact-subset-0094.json)