R0046-23580 · q-20261014-advance-20261112-r03-compact-subset-closed-in-hausdorff-0080
Original question and givens
Use complete finite labeled carrier=[o4, u4], topology=[[o4, u4], [u4], []] in its original supplied open-set order, and compact_subset K=[u4]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
G001 — Use complete finite labeled carrier=[o4, u4], topology=[[o4, u4], [u4], []] in its original supplied open-set order, and compact_subset K=[u4]. Qualify the topology and every Hausdorff distinct-point pair. Finite K is compact; the general theorem still needs both ambient Hausdorffness and compactness. Give (points, compact_subset, complement, Hausdorff, compact, closed, boundary).
Step-by-step derivation
S001 · implies
Before
source operation=compactclosed-nonhausdorff-boundary;original snapshot and mathematics retained
After
explicit ordered source output fields=[points, compact_subset, complement, Hausdorff, compact, closed, boundary]
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=[o4, u4];ORIGINAL open-list order=[[o4, u4], [u4], []];point=o4;ALL given sets containing it have indices=[0]
After
candidate N(o4)=intersection of ALL those sets=[o4, u4];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=[o4, u4];ORIGINAL open-list order=[[o4, u4], [u4], []];point=u4;ALL given sets containing it have indices=[0, 1]
After
candidate N(u4)=intersection of ALL those sets=[u4];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 ordered open masks=[3, 2, 0];ALL neighborhood-containment [x,y,N(y)_subset_N(x)] rows=[[o4, o4, Yes], [o4, u4, Yes], [u4, u4, Yes]]
After
ALL carrier masks0..3 audited;EXACT qualifying U with every x in U having N(x) subset U=[0, 2, 3];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, S003
S005 · implies
Before
Compact-closed ORIGINAL pair index0-1:ORIGINAL distinct x=o4,y=u4;N(y)=[u4];open-list order=[[o4, u4], [u4], []]
After
ALL tested [U_index,U_intersect_N(y),disjoint_V_possible] rows=[[0, [u4], No]];for FIRST possible U,ALL tested [V_index,intersection,disjoint] rows=[];FIRST ORIGINAL witness=null
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, S004
S006 · implies
Before
ORIGINAL points=[o4, u4];ALL original-index distinct-pair witnesses=[[o4, u4, null]];actual K=[u4]
After
Hausdorff=No;K compact=Yes by finiteness:from ANY cover choose one member per K point;empty K needs no members;ambient complement=[o4],closed=No
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, S005
Result
([o4, u4], [u4], [o4], No, Yes, No, Sierpinski_type_space_shows_Hausdorff_hypothesis_cannot_be_removed_and_converse_is_not_claimed)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.