Question 9755: Neighborhood Upward-Closure Guards

Original question and givens

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 1, 2], [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, small=[0], 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=[[], [0], [0, 2], [0, 1, 2], [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, small=[0], 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.

• mko-neighborhood

Step-by-step derivation

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

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

point candidate=-1;underlying Fin(4)

point qualified=No

raw small=[0];raw large=[0, 1, 2, 3]

small=[0];large=[0, 1, 2, 3];small subset large=Yes

point qualified=No;included=Yes;small neighborhood/status=not evaluated

conditional large neighborhood/status=not evaluated

Reverse checks

• 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, No, [0], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated)

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.