Question 9821: Complete Open, Closed, and Clopen Families

Original question and givens

Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [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. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.

• G001 — Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [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. Give (n, all opens, all closed sets, all clopen sets), 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.

• mko-open-closed-set

Step-by-step derivation

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

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

universe=[0, 1, 2, 3];open=[]

closed relative complement=[0, 1, 2, 3]

universe=[0, 1, 2, 3];open=[0]

closed relative complement=[1, 2, 3]

universe=[0, 1, 2, 3];open=[0, 3]

closed relative complement=[1, 2]

universe=[0, 1, 2, 3];open=[0, 2, 3]

closed relative complement=[1]

universe=[0, 1, 2, 3];open=[0, 1, 2, 3]

closed relative complement=[]

all opens=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]];all closed=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]

all clopen=[[], [0, 1, 2, 3]]

Reverse checks

• 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, [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [1, 2, 3], [0, 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.