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Question 22902: All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0044-22902 · q-20261014-advance-20261111-r03-hausdorff-space-0027

Original question and givens

Use finite unique labeled points=[a1, b1, c1] and the complete canonical topology=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

  • G001 — Use finite unique labeled points=[a1, b1, c1] and the complete canonical topology=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, 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.

Step-by-step derivation

S001 · implies

Before

source operation=hausdorff-full-audit;original snapshot and mathematics retained

After

explicit ordered source output fields=[points, pair_witness_rows, Hausdorff, certificate]

Conditions: none

Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=0,U=[a1, b1, c1]

After

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [a1, b1, c1]], [1, [a1, b1, c1], [a1, b1]], [2, [a1, b1, c1], [a1, c1]], [3, [a1, b1, c1], [a1]], [4, [a1, b1, c1], [b1, c1]], [5, [a1, b1, c1], [b1]], [6, [a1, b1, c1], [c1]], [7, [a1, b1, c1], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S001

S003 · implies

Before

ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=1,U=[a1, b1]

After

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [a1, b1]], [1, [a1, b1], [a1, b1]], [2, [a1, b1, c1], [a1]], [3, [a1, b1], [a1]], [4, [a1, b1, c1], [b1]], [5, [a1, b1], [b1]], [6, [a1, b1, c1], []], [7, [a1, b1], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S002

S004 · implies

Before

ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=2,U=[a1, c1]

After

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [a1, c1]], [1, [a1, b1, c1], [a1]], [2, [a1, c1], [a1, c1]], [3, [a1, c1], [a1]], [4, [a1, b1, c1], [c1]], [5, [a1, b1, c1], []], [6, [a1, c1], [c1]], [7, [a1, c1], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S003

S005 · implies

Before

ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=3,U=[a1]

After

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [a1]], [1, [a1, b1], [a1]], [2, [a1, c1], [a1]], [3, [a1], [a1]], [4, [a1, b1, c1], []], [5, [a1, b1], []], [6, [a1, c1], []], [7, [a1], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S004

S006 · implies

Before

ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=4,U=[b1, c1]

After

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [b1, c1]], [1, [a1, b1, c1], [b1]], [2, [a1, b1, c1], [c1]], [3, [a1, b1, c1], []], [4, [b1, c1], [b1, c1]], [5, [b1, c1], [b1]], [6, [b1, c1], [c1]], [7, [b1, c1], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S005

S007 · implies

Before

ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=5,U=[b1]

After

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [b1]], [1, [a1, b1], [b1]], [2, [a1, b1, c1], []], [3, [a1, b1], []], [4, [b1, c1], [b1]], [5, [b1], [b1]], [6, [b1, c1], []], [7, [b1], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S006

S008 · implies

Before

ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=6,U=[c1]

After

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [c1]], [1, [a1, b1, c1], []], [2, [a1, c1], [c1]], [3, [a1, c1], []], [4, [b1, c1], [c1]], [5, [b1, c1], []], [6, [c1], [c1]], [7, [c1], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S007

S009 · implies

Before

ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []];actual open index=7,U=[]

After

ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], []], [1, [a1, b1], []], [2, [a1, c1], []], [3, [a1], []], [4, [b1, c1], []], [5, [b1], []], [6, [c1], []], [7, [], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S008

S010 · implies

Before

Hausdorff ORIGINAL pair index0:ORIGINAL distinct carrier points x=a1,y=b1;source open order=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []]

After

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a1, b1, c1], No], [0, 1, [a1, b1], No], [0, 4, [b1, c1], No], [0, 5, [b1], No], [1, 0, [a1, b1], No], [1, 1, [a1, b1], No], [1, 4, [b1], No], [1, 5, [b1], No], [2, 0, [a1, c1], No], [2, 1, [a1], No], [2, 4, [c1], No], [2, 5, [], Yes]];FIRST source-order witness=[[a1, c1], [b1]]

Conditions: none

Use actual opens U containing x and V containing y.Search original U then V order for the first disjoint pair.Null means no witness in this complete finite domain,not a failed sample.

Step identifiers and dependencies

Rule: hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure · Uses: G001, S009

S011 · implies

Before

Hausdorff ORIGINAL pair index1:ORIGINAL distinct carrier points x=a1,y=c1;source open order=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []]

After

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a1, b1, c1], No], [0, 2, [a1, c1], No], [0, 4, [b1, c1], No], [0, 6, [c1], No], [1, 0, [a1, b1], No], [1, 2, [a1], No], [1, 4, [b1], No], [1, 6, [], Yes]];FIRST source-order witness=[[a1, b1], [c1]]

Conditions: none

Use actual opens U containing x and V containing y.Search original U then V order for the first disjoint pair.Null means no witness in this complete finite domain,not a failed sample.

Step identifiers and dependencies

Rule: hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure · Uses: G001, S010

S012 · implies

Before

Hausdorff ORIGINAL pair index2:ORIGINAL distinct carrier points x=b1,y=c1;source open order=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [b1, c1], [b1], [c1], []]

After

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a1, b1, c1], No], [0, 2, [a1, c1], No], [0, 4, [b1, c1], No], [0, 6, [c1], No], [1, 0, [a1, b1], No], [1, 2, [a1], No], [1, 4, [b1], No], [1, 6, [], Yes]];FIRST source-order witness=[[a1, b1], [c1]]

Conditions: none

Use actual opens U containing x and V containing y.Search original U then V order for the first disjoint pair.Null means no witness in this complete finite domain,not a failed sample.

Step identifiers and dependencies

Rule: hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure · Uses: G001, S011

S013 · implies

Before

ORIGINAL carrier order=[a1, b1, c1];ALL unordered distinct pairs with first index<second=[[a1, b1], [a1, c1], [b1, c1]]

After

ALL pair-witness rows=[[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]];Hausdorff=Yes;no pairs means vacuous True,not failure

Conditions: none

One exhaustively failed pair refutes Hausdorffness;all pairs must succeed for this finite certificate.For a finite Hausdorff space,intersect x-neighborhoods excluding each other y to obtain open singleton{x},hence discreteness.Do not extrapolate that corollary to infinite spaces.

Step identifiers and dependencies

Rule: hausdorff_ALL_original_point_pairs_empty_singleton_vacuity · 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

([a1, b1, c1], [[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]], Yes, all_distinct_pairs_exhaustively_separated)

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: 8eb5efbb83df3442b3d782dbc5ed33f24934a5fc08bfd40abba58f5845ab392f
  • candidate record SHA-256: 7a72801f3f6e26cdb2486d03e58bee407a7c99acda61438dde352be658fa6a4c
  • semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
  • question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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