Question 22906: All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

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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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R0044-22906 · q-20261014-advance-20261111-r03-hausdorff-space-0031

Original question and givens

Use finite unique labeled points=[a5, b5, c5] and the complete canonical topology=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []]. 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=[a5, b5, c5] and the complete canonical topology=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []]. 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.

mko-hausdorff-space

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=[a5, b5, c5];source canonical open family=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []];actual open index=0,U=[a5, b5, c5]

After

ALL [other_index,union,intersection] rows=[[0, [a5, b5, c5], [a5, b5, c5]], [1, [a5, b5, c5], [a5, b5]], [2, [a5, b5, c5], [a5, c5]], [3, [a5, b5, c5], [a5]], [4, [a5, b5, c5], [b5, c5]], [5, [a5, b5, c5], [b5]], [6, [a5, b5, c5], [c5]], [7, [a5, b5, c5], []]];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=[a5, b5, c5];source canonical open family=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []];actual open index=1,U=[a5, b5]

After

ALL [other_index,union,intersection] rows=[[0, [a5, b5, c5], [a5, b5]], [1, [a5, b5], [a5, b5]], [2, [a5, b5, c5], [a5]], [3, [a5, b5], [a5]], [4, [a5, b5, c5], [b5]], [5, [a5, b5], [b5]], [6, [a5, b5, c5], []], [7, [a5, b5], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S002

S004 · implies

Before

ORIGINAL carrier order=[a5, b5, c5];source canonical open family=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []];actual open index=2,U=[a5, c5]

After

ALL [other_index,union,intersection] rows=[[0, [a5, b5, c5], [a5, c5]], [1, [a5, b5, c5], [a5]], [2, [a5, c5], [a5, c5]], [3, [a5, c5], [a5]], [4, [a5, b5, c5], [c5]], [5, [a5, b5, c5], []], [6, [a5, c5], [c5]], [7, [a5, c5], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S003

S005 · implies

Before

ORIGINAL carrier order=[a5, b5, c5];source canonical open family=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []];actual open index=3,U=[a5]

After

ALL [other_index,union,intersection] rows=[[0, [a5, b5, c5], [a5]], [1, [a5, b5], [a5]], [2, [a5, c5], [a5]], [3, [a5], [a5]], [4, [a5, b5, c5], []], [5, [a5, b5], []], [6, [a5, c5], []], [7, [a5], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S004

S006 · implies

Before

ORIGINAL carrier order=[a5, b5, c5];source canonical open family=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []];actual open index=4,U=[b5, c5]

After

ALL [other_index,union,intersection] rows=[[0, [a5, b5, c5], [b5, c5]], [1, [a5, b5, c5], [b5]], [2, [a5, b5, c5], [c5]], [3, [a5, b5, c5], []], [4, [b5, c5], [b5, c5]], [5, [b5, c5], [b5]], [6, [b5, c5], [c5]], [7, [b5, c5], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S005

S007 · implies

Before

ORIGINAL carrier order=[a5, b5, c5];source canonical open family=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []];actual open index=5,U=[b5]

After

ALL [other_index,union,intersection] rows=[[0, [a5, b5, c5], [b5]], [1, [a5, b5], [b5]], [2, [a5, b5, c5], []], [3, [a5, b5], []], [4, [b5, c5], [b5]], [5, [b5], [b5]], [6, [b5, c5], []], [7, [b5], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S006

S008 · implies

Before

ORIGINAL carrier order=[a5, b5, c5];source canonical open family=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []];actual open index=6,U=[c5]

After

ALL [other_index,union,intersection] rows=[[0, [a5, b5, c5], [c5]], [1, [a5, b5, c5], []], [2, [a5, c5], [c5]], [3, [a5, c5], []], [4, [b5, c5], [c5]], [5, [b5, c5], []], [6, [c5], [c5]], [7, [c5], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S007

S009 · implies

Before

ORIGINAL carrier order=[a5, b5, c5];source canonical open family=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []];actual open index=7,U=[]

After

ALL [other_index,union,intersection] rows=[[0, [a5, b5, c5], []], [1, [a5, b5], []], [2, [a5, c5], []], [3, [a5], []], [4, [b5, c5], []], [5, [b5], []], [6, [c5], []], [7, [], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

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=a5,y=b5;source open order=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []]

After

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a5, b5, c5], No], [0, 1, [a5, b5], No], [0, 4, [b5, c5], No], [0, 5, [b5], No], [1, 0, [a5, b5], No], [1, 1, [a5, b5], No], [1, 4, [b5], No], [1, 5, [b5], No], [2, 0, [a5, c5], No], [2, 1, [a5], No], [2, 4, [c5], No], [2, 5, [], Yes]];FIRST source-order witness=[[a5, c5], [b5]]

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=a5,y=c5;source open order=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []]

After

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a5, b5, c5], No], [0, 2, [a5, c5], No], [0, 4, [b5, c5], No], [0, 6, [c5], No], [1, 0, [a5, b5], No], [1, 2, [a5], No], [1, 4, [b5], No], [1, 6, [], Yes]];FIRST source-order witness=[[a5, b5], [c5]]

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=b5,y=c5;source open order=[[a5, b5, c5], [a5, b5], [a5, c5], [a5], [b5, c5], [b5], [c5], []]

After

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a5, b5, c5], No], [0, 2, [a5, c5], No], [0, 4, [b5, c5], No], [0, 6, [c5], No], [1, 0, [a5, b5], No], [1, 2, [a5], No], [1, 4, [b5], No], [1, 6, [], Yes]];FIRST source-order witness=[[a5, b5], [c5]]

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=[a5, b5, c5];ALL unordered distinct pairs with first index<second=[[a5, b5], [a5, c5], [b5, c5]]

After

ALL pair-witness rows=[[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]];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

([a5, b5, c5], [[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]], 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: 43abb26c4d8b68e1da69bfa78255faef1f0a9bc623b2a44236f30546d43b8cf4
candidate record SHA-256: c638c4a83de4b082da46646d57f36a50a183d2bc02933146dcfc5eb3f41f0199
semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
Lean: NOT_RUN · human review: NOT_PERFORMED

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