Use the complete finite topology on X=Fin(3)={0,…,2} with opens=[[], [0], [0, 1], [0, 1, 2]]. 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. mode=open_union, selected=[[], [0, 1]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.
G001 — Use the complete finite topology on X=Fin(3)={0,…,2} with opens=[[], [0], [0, 1], [0, 1, 2]]. 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. mode=open_union, selected=[[], [0, 1]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.
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.
The empty union is empty and the empty intersection is the same whole universe.Start from that identity and fold the selected sets.The all-qualified condition is true for an empty family.
all selected properties qualified=Yes;result=[0, 1]
After
result property qualified=Yes
Conditions: none
Qualified premises guarantee the corresponding closure property.If premises fail,the result may still qualify accidentally,so compute the two fields separately.
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
(3, open_union, [[], [0, 1]], Yes, [0, 1], 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.