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Question 9722: All Neighborhoods of a Qualified Finite Point

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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R0018-09722 · q-20261013-advance-20261019-r02-neighborhood-0097

Original question and givens

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=1 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.

  • G001 — Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=1 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.

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

n=4;raw open family=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

universe=[0, 1, 2, 3];normalized opens=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]];empty=Yes;full=Yes;union closed=Yes;intersection closed=Yes

Conditions: none

Deduplicate subsets/families and order by masks.Check empty/whole membership and all binary union/intersection closure.On a finite underlying set,arbitrary unions reduce to finitely many distinct members.These tasks require an actual topology;invalid data is not used for later properties.

Step identifiers and dependencies

Rule: given_complete_finite_topology_guard · Uses: G001

S002 · implies

Before

point candidate=1;underlying Fin(4)

After

point qualified=Yes

Conditions: none

An outside point leaves neighborhood claims unevaluated,not False via negative indexing or an empty witness list.Candidate sets must still independently be typed.

Step identifiers and dependencies

Rule: neighborhood_point_domain_guard · Uses: G001, S001

S003 · implies

Before

point=1;complete open family=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all point-containing opens=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

Conditions: none

Take every open containing the qualified point as the complete witness candidates.A neighborhood itself need not be open;it must contain at least one of them.

Step identifiers and dependencies

Rule: complete_point_open_witness_basis · Uses: G001, S002

S004 · implies

Before

phase=powerset mask0;point=1;candidate=[];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[];neighborhood=No

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S003

S005 · implies

Before

phase=powerset mask1;point=1;candidate=[0];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[];neighborhood=No

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S004

S006 · implies

Before

phase=powerset mask2;point=1;candidate=[1];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[[1]];neighborhood=Yes

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S005

S007 · implies

Before

phase=powerset mask3;point=1;candidate=[0, 1];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[[1]];neighborhood=Yes

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S006

S008 · implies

Before

phase=powerset mask4;point=1;candidate=[2];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[];neighborhood=No

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S007

S009 · implies

Before

phase=powerset mask5;point=1;candidate=[0, 2];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[];neighborhood=No

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S008

S010 · implies

Before

phase=powerset mask6;point=1;candidate=[1, 2];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[[1]];neighborhood=Yes

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S009

S011 · implies

Before

phase=powerset mask7;point=1;candidate=[0, 1, 2];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[[1]];neighborhood=Yes

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S010

S012 · implies

Before

phase=powerset mask8;point=1;candidate=[3];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[];neighborhood=No

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S011

S013 · implies

Before

phase=powerset mask9;point=1;candidate=[0, 3];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[];neighborhood=No

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S012

S014 · implies

Before

phase=powerset mask10;point=1;candidate=[1, 3];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[[1], [1, 3]];neighborhood=Yes

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S013

S015 · implies

Before

phase=powerset mask11;point=1;candidate=[0, 1, 3];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[[1], [1, 3]];neighborhood=Yes

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S014

S016 · implies

Before

phase=powerset mask12;point=1;candidate=[2, 3];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[];neighborhood=No

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S015

S017 · implies

Before

phase=powerset mask13;point=1;candidate=[0, 2, 3];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[];neighborhood=No

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S016

S018 · implies

Before

phase=powerset mask14;point=1;candidate=[1, 2, 3];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[[1], [1, 3], [1, 2, 3]];neighborhood=Yes

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S017

S019 · implies

Before

phase=powerset mask15;point=1;candidate=[0, 1, 2, 3];point-open basis=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]

After

all contained open witnesses=[[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]];neighborhood=Yes

Conditions: none

List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership.

Step identifiers and dependencies

Rule: all_open_neighborhood_witnesses · Uses: G001, S018

Reverse checks

  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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

(4, 1, [[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])

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: 58d04c777fa61a7f95ebf599dd7ecf6d94b1cdec8adc244c9b63e02859294abe
  • candidate record SHA-256: 5df0974a78974d3f117218154f7a46d4c78abd6b0ac38d3260d4c236935eb537
  • semantic review receipt SHA-256: bd35da62087a15d6b89b45585dc46e89b9efbe097cba7b3c707b5030496f013c
  • question-first witness SHA-256: fab45a10d8ac95c7d61d32e47d1bdd3ca862c17c3289a6bf4b01d4dc9492fbb6
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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