R0046-23574 · q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0074
Step-by-step derivation
S001 · implies
Before
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 dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL carrier order=[a23, b23, c23];ORIGINAL open-list order=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []];point=a23;ALL given sets containing it have indices=[0, 1, 2, 3]
After
candidate N(a23)=intersection of ALL those sets=[a23];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 dependencies
Rule: compactclosed_ALL_given_neighborhood_intersection_candidates · Uses: G001, S001
S003 · implies
Before
ORIGINAL carrier order=[a23, b23, c23];ORIGINAL open-list order=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []];point=b23;ALL given sets containing it have indices=[0, 1, 4, 5]
After
candidate N(b23)=intersection of ALL those sets=[b23];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 dependencies
Rule: compactclosed_ALL_given_neighborhood_intersection_candidates · Uses: G001, S002
S004 · implies
Before
ORIGINAL carrier order=[a23, b23, c23];ORIGINAL open-list order=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []];point=c23;ALL given sets containing it have indices=[0, 2, 4, 6]
After
candidate N(c23)=intersection of ALL those sets=[c23];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 dependencies
Rule: compactclosed_ALL_given_neighborhood_intersection_candidates · Uses: G001, S003
S005 · implies
Before
ORIGINAL ordered open masks=[7, 3, 5, 1, 6, 2, 4, 0];ALL neighborhood-containment [x,y,N(y)_subset_N(x)] rows=[[a23, a23, Yes], [b23, b23, Yes], [c23, c23, 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 dependencies
Rule: compactclosed_ALL_carrier_subsets_exact_neighborhood_upsets_topology · Uses: G001, S004
S006 · implies
Before
Compact-closed ORIGINAL pair index0-1:ORIGINAL distinct x=a23,y=b23;N(y)=[b23];open-list order=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []]
After
ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [b23], No], [1, [b23], No], [2, [], Yes]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a23, c23], No], [1, [a23], No], [4, [c23], No], [5, [], Yes]];FIRST ORIGINAL witness=[[a23, c23], [b23]]
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 dependencies
Rule: compactclosed_FIRST_original_separator_with_complete_neighborhood_screen · Uses: G001, S005
S007 · implies
Before
Compact-closed ORIGINAL pair index0-2:ORIGINAL distinct x=a23,y=c23;N(y)=[c23];open-list order=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []]
After
ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [c23], No], [1, [], Yes]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a23, b23], No], [2, [a23], No], [4, [b23], No], [6, [], Yes]];FIRST ORIGINAL witness=[[a23, b23], [c23]]
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 dependencies
Rule: compactclosed_FIRST_original_separator_with_complete_neighborhood_screen · Uses: G001, S006
S008 · implies
Before
Compact-closed ORIGINAL pair index1-2:ORIGINAL distinct x=b23,y=c23;N(y)=[c23];open-list order=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []]
After
ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [c23], No], [1, [], Yes]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a23, b23], No], [2, [a23], No], [4, [b23], No], [6, [], Yes]];FIRST ORIGINAL witness=[[a23, b23], [c23]]
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 dependencies
Rule: compactclosed_FIRST_original_separator_with_complete_neighborhood_screen · Uses: G001, S007
S009 · implies
Before
ORIGINAL points=[a23, b23, c23];ALL original-index distinct-pair witnesses=[[a23, b23, [[a23, c23], [b23]]], [a23, c23, [[a23, b23], [c23]]], [b23, c23, [[a23, b23], [c23]]]];actual K=[a23, b23]
After
Hausdorff=Yes;K compact=Yes by finiteness:from ANY cover choose one member per K point;empty K needs no members;ambient complement=[c23],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 dependencies
Rule: compactclosed_ALL_pairs_Hausdorff_and_finite_subset_compact_separate · Uses: G001, S008
S010 · implies
Before
CURRENT ambient Hausdorff=Yes;compact K=[a23, b23];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 dependencies
Rule: compactclosed_GENERAL_Hausdorff_finite_subcover_intersection_proof · Uses: G001, S009
S011 · implies
Before
Compact-closed ORIGINAL exterior versus K index0:ORIGINAL distinct x=c23,y=a23;N(y)=[a23];open-list order=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []]
After
ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [a23], No], [2, [a23], No], [4, [], Yes]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [b23, c23], No], [1, [b23], No], [2, [c23], No], [3, [], Yes]];FIRST ORIGINAL witness=[[b23, c23], [a23]]
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 dependencies
Rule: compactclosed_FIRST_original_separator_with_complete_neighborhood_screen · Uses: G001, S010
S012 · implies
Before
Compact-closed ORIGINAL exterior versus K index1:ORIGINAL distinct x=c23,y=b23;N(y)=[b23];open-list order=[[a23, b23, c23], [a23, b23], [a23, c23], [a23], [b23, c23], [b23], [c23], []]
After
ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [b23], No], [2, [], Yes]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[[0, [a23, c23], No], [1, [a23], No], [4, [c23], No], [5, [], Yes]];FIRST ORIGINAL witness=[[a23, c23], [b23]]
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 dependencies
Rule: compactclosed_FIRST_original_separator_with_complete_neighborhood_screen · Uses: G001, S011
S013 · implies
Before
ORIGINAL exterior x=c23;sorted K=[a23, b23];ALL [y,FIRST original separator]= [[a23, [[b23, c23], [a23]]], [b23, [[a23, c23], [b23]]]]
After
finite intersection of U sides=[c23];union of V sides=[a23, b23];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 dependencies
Rule: compactclosed_ACTUAL_cover_union_finite_U_intersection_original_order · Uses: G001, S012