Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 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=3, small=[3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
G001 — Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 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=3, small=[3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).
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.
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.
An outside point leaves neighborhood claims unevaluated,not False via negative indexing or an empty witness list.Candidate sets must still independently be typed.
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.
all contained open witnesses=[[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.
point qualified=Yes;included=Yes;small neighborhood/status=Yes
After
conditional large neighborhood/status=Yes
Conditions: none
Evaluate the last field only after every specified premise holds.A failed premise leaves it unevaluated rather than proving large is not a neighborhood.
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, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)
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.