# Question 23563: Finite Intersections Separate an Exterior Point from the Whole Compact Subset

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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.

R0046-23563 · q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0063



## Original question and givens

Use complete finite labeled carrier=[a12, b12, c12], topology=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []] in its original supplied open-set order, and compact_subset K=[a12, b12]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c12 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).


- G001 — Use complete finite labeled carrier=[a12, b12, c12], topology=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []] in its original supplied open-set order, and compact_subset K=[a12, b12]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Exterior point x=c12 must lie outside K. Give (exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate).





## 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-closed-in-hausdorff](https://ocme.evemisslab.com/en/mko/mko-compact-subset-closed-in-hausdorff/)





## Step-by-step derivation

S001 · impliesBefore

source operation=compactclosed-neighborhood-subcover-certificate;original snapshot and mathematics retained

After

explicit ordered source output fields=[exterior_point, separation_rows, exterior_neighborhood, cover_union, covers_K, neighborhood_disjoint_from_K, neighborhood_open, certificate]

Conditions: none

Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ORIGINAL carrier order=[a12, b12, c12];ORIGINAL open-list order=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []];point=a12;ALL given sets containing it have indices=[0, 1, 2, 3]

After

candidate N(a12)=intersection of ALL those sets=[a12];no original open-list reordering

Conditions: none

Intersect every declared neighborhood to form a candidate,without yet assuming a topology.Retain the original open-list order;only set elements use UTF16 sorting.

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





S003 · impliesBefore

ORIGINAL carrier order=[a12, b12, c12];ORIGINAL open-list order=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []];point=b12;ALL given sets containing it have indices=[0, 1, 4, 5]

After

candidate N(b12)=intersection of ALL those sets=[b12];no original open-list reordering

Conditions: none

Intersect every declared neighborhood to form a candidate,without yet assuming a topology.Retain the original open-list order;only set elements use UTF16 sorting.

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





S004 · impliesBefore

ORIGINAL carrier order=[a12, b12, c12];ORIGINAL open-list order=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []];point=c12;ALL given sets containing it have indices=[0, 2, 4, 6]

After

candidate N(c12)=intersection of ALL those sets=[c12];no original open-list reordering

Conditions: none

Intersect every declared neighborhood to form a candidate,without yet assuming a topology.Retain the original open-list order;only set elements use UTF16 sorting.

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





S005 · impliesBefore

ORIGINAL ordered open masks=[7, 3, 5, 1, 6, 2, 4, 0];ALL neighborhood-containment [x,y,N(y)_subset_N(x)] rows=[[a12, a12, Yes], [b12, b12, Yes], [c12, c12, Yes]]

After

ALL carrier masks0..7 audited;EXACT qualifying U with every x in U having N(x) subset U=[0, 1, 2, 3, 4, 5, 6, 7];given family equals this COMPLETE upset family

Conditions: none

In a finite topology,opens are exactly upsets of its minimal neighborhoods.Conversely,after comparing every carrier subset,upsets are closed under arbitrary unions and finite intersections and contain empty/full sets,so this is a topology with open N(x).This covers all closure obligations,not sampled pairs,without printing millions of pairs.

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





S006 · impliesBefore

Compact-closed ORIGINAL pair index0-1:ORIGINAL distinct x=a12,y=b12;N(y)=[b12];open-list order=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []]

After

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [b12], No], [1, [b12], No], [2, [], Yes]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a12, c12], No], [1, [a12], No], [4, [c12], No], [5, [], Yes]];FIRST ORIGINAL witness=[[a12, c12], [b12]]

Conditions: none

If U meets N(y),every open containing y also meets U.If disjoint,N(y) itself is an available open,then search original V order for the first witness.This preserves exactly the original full U/V search result,not an alternate minimal-neighborhood answer.

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





S007 · impliesBefore

Compact-closed ORIGINAL pair index0-2:ORIGINAL distinct x=a12,y=c12;N(y)=[c12];open-list order=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []]

After

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [c12], No], [1, [], Yes]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a12, b12], No], [2, [a12], No], [4, [b12], No], [6, [], Yes]];FIRST ORIGINAL witness=[[a12, b12], [c12]]

Conditions: none

If U meets N(y),every open containing y also meets U.If disjoint,N(y) itself is an available open,then search original V order for the first witness.This preserves exactly the original full U/V search result,not an alternate minimal-neighborhood answer.

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





S008 · impliesBefore

Compact-closed ORIGINAL pair index1-2:ORIGINAL distinct x=b12,y=c12;N(y)=[c12];open-list order=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []]

After

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [c12], No], [1, [], Yes]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a12, b12], No], [2, [a12], No], [4, [b12], No], [6, [], Yes]];FIRST ORIGINAL witness=[[a12, b12], [c12]]

Conditions: none

If U meets N(y),every open containing y also meets U.If disjoint,N(y) itself is an available open,then search original V order for the first witness.This preserves exactly the original full U/V search result,not an alternate minimal-neighborhood answer.

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





S009 · impliesBefore

ORIGINAL points=[a12, b12, c12];ALL original-index distinct-pair witnesses=[[a12, b12, [[a12, c12], [b12]]], [a12, c12, [[a12, b12], [c12]]], [b12, c12, [[a12, b12], [c12]]]];actual K=[a12, b12]

After

Hausdorff=Yes;K compact=Yes by finiteness:from ANY cover choose one member per K point;empty K needs no members;ambient complement=[c12],closed=Yes

Conditions: none

Finite K compactness does not replace ambient Hausdorffness.An empty pair domain is vacuous.Closedness tests the complement in this same ambient space,not a relative complement.

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





S010 · impliesBefore

CURRENT ambient Hausdorff=Yes;compact K=[a12, b12];consider ANY exterior x

After

for EACH y in K choose disjoint opens U_y containing x,V_y containing y;the V_y cover K;compactness picks FINITELY MANY indices;their U_y intersection is open,contains x and avoids ALL K,so X minus K is open

Conditions: none

The general theorem uses compactness to select a finite subcover,not numbers from this finite model.Each K point lies in a selected V disjoint from its corresponding U;the finite intersection therefore avoids all K.For empty K,the empty intersection is X.No converse closed-implies-compact is claimed.

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





S011 · impliesBefore

Compact-closed ORIGINAL exterior versus K index0:ORIGINAL distinct x=c12,y=a12;N(y)=[a12];open-list order=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []]

After

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [a12], No], [2, [a12], No], [4, [], Yes]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [b12, c12], No], [1, [b12], No], [2, [c12], No], [3, [], Yes]];FIRST ORIGINAL witness=[[b12, c12], [a12]]

Conditions: none

If U meets N(y),every open containing y also meets U.If disjoint,N(y) itself is an available open,then search original V order for the first witness.This preserves exactly the original full U/V search result,not an alternate minimal-neighborhood answer.

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





S012 · impliesBefore

Compact-closed ORIGINAL exterior versus K index1:ORIGINAL distinct x=c12,y=b12;N(y)=[b12];open-list order=[[a12, b12, c12], [a12, b12], [a12, c12], [a12], [b12, c12], [b12], [c12], []]

After

ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [b12], No], [2, [], Yes]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a12, c12], No], [1, [a12], No], [4, [c12], No], [5, [], Yes]];FIRST ORIGINAL witness=[[a12, c12], [b12]]

Conditions: none

If U meets N(y),every open containing y also meets U.If disjoint,N(y) itself is an available open,then search original V order for the first witness.This preserves exactly the original full U/V search result,not an alternate minimal-neighborhood answer.

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





S013 · impliesBefore

ORIGINAL exterior x=c12;sorted K=[a12, b12];ALL [y,FIRST original separator]= [[a12, [[b12, c12], [a12]]], [b12, [[a12, c12], [b12]]]]

After

finite intersection of U sides=[c12];union of V sides=[a12, b12];covers K=Yes;intersection avoids K=Yes;intersection open=Yes

Conditions: none

The V union may exceed K while still covering it.The U intersection must be open,contain x and avoid K.No minimum/unique subcover is claimed;for empty K retain the raw carrier order in the empty intersection.

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









## Reverse checks


- 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

(c12, [[a12, [[b12, c12], [a12]]], [b12, [[a12, c12], [b12]]]], [c12], [a12, b12], Yes, Yes, Yes, finite_subcover_and_finite_intersection_separate_x_from_K)

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: eae96a7c95a2d535a39d31adb440aecaa9fdf77cbef4b021523c38b9e1f1cf37
- candidate record SHA-256: 9509d4867745664b44152692f384792829d0401d01bdca5b4c61fed17102adf0
- semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
- question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
- Lean: NOT_RUN · human review: NOT_PERFORMED

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