Question 21656: Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0042-21656 · q-20261014-advance-20261109-r03-compact-subset-0031

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]], subset K=[1, 2, 3] 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. Supplied indexed cover=[[], [1], [1, 2, 3], [1, 3]]. Give (cover, covers_subset, union, 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]], subset K=[1, 2, 3] 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. Supplied indexed cover=[[], [1], [1, 2, 3], [1, 3]]. Give (cover, covers_subset, union, 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

Step-by-step derivation

S001 · implies

Before

source operation=compact-open-cover-audit;original snapshot and mathematics retained

After

explicit ordered source output fields=[cover, covers_subset, union, 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 dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

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 dependencies

Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S001

S003 · implies

Before

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

Step identifiers and dependencies

Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S002

S004 · implies

Before

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

Step identifiers and dependencies

Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S003

S005 · implies

Before

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

Step identifiers and dependencies

Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S004

S006 · implies

Before

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

Step identifiers and dependencies

Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S005

S007 · implies

Before

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

Step identifiers and dependencies

Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S006

S008 · implies

Before

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 dependencies

Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S007

S009 · implies

Before

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 dependencies

Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S008

S010 · implies

Before

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 dependencies

Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S009

S011 · implies

Before

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 dependencies

Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S010

S012 · implies

Before

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 dependencies

Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S011

S013 · implies

Before

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 dependencies

Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S012

S014 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

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 dependencies

Rule: subtop_induced_axioms_and_empty_Y · Uses: G001, S014

S016 · implies

Before

ORIGINAL K=[1, 2, 3];COMPLETE finite relative open family=[[], [1], [1, 2, 3], [1, 3]];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 dependencies

Rule: compact_finite_DISTINCT_open_family_general_cover_argument · Uses: G001, S015

S017 · implies

Before

ORIGINAL indexed cover=[[], [1], [1, 2, 3], [1, 3]];relative-open family=[[], [1], [1, 2, 3], [1, 3]];K=[1, 2, 3]

After

ALL supplied members are relative-open;full union=[1, 2, 3];covers K=Yes;outer duplicate positions retained

Conditions: none

Audit each member as relative-open and compute the full union.Retain duplicate outer positions as distinct original indices.One successful cover is not a general compactness proof;a failing list reports false.

Step identifiers and dependencies

Rule: compact_actual_indexed_cover_relative_opens_and_union · 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], [1, 2, 3], [1, 3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)

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: dbfe7e09f9b8854c493ac742141832cfda690a08e8f8e604352e57842a8a39a5
candidate record SHA-256: edbd9f8db56d17e8004a5706aa33347732c1330010218cb3a3b0231c5cbd7b5e
semantic review receipt SHA-256: dd44555041dfbb6fe9936aa00b9a5f8c5982c2784ef61dd0cbd5d1a303a49c1a
question-first witness SHA-256: 9a6011263f6ac53b9a1fd57addf2f3e2f6088b54802c539f332d689480dc2d17
Lean: NOT_RUN · human review: NOT_PERFORMED

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