# Question 9865: Empty Families and Qualified Closure Premises [Return to catalogue page 50](https://ocme.evemisslab.com/en/questions/page/50/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261019-r01-open-closed-set-foundations/#q-20261013-advance-20261019-r01-open-closed-set-0090) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0018-09865 · q-20261013-advance-20261019-r01-open-closed-set-0090 ## Original question and givens Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [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. mode=closed_intersection, selected=[[], [1, 3], [0, 1, 2, 3]]. 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(4)={0,…,3} with opens=[[], [2], [0, 2], [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. mode=closed_intersection, selected=[[], [1, 3], [0, 1, 2, 3]]. 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. - [mko-open-closed-set](https://ocme.evemisslab.com/en/mko/mko-open-closed-set/) ## Step-by-step derivation S001 · impliesBefore n=4;raw open family=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]] After universe=[0, 1, 2, 3];normalized opens=[[], [2], [0, 2], [0, 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 dependenciesRule: given_complete_finite_topology_guard · Uses: G001 S002 · impliesBefore mode=closed_intersection;selected member=[] After required property qualified=Yes Conditions: none Audit the required open/closed property of every selected set by the chosen mode.A failure does not stop computation of the union/intersection. Step identifiers and dependenciesRule: closure_each_selected_property · Uses: G001, S001 S003 · impliesBefore mode=closed_intersection;selected member=[1, 3] After required property qualified=Yes Conditions: none Audit the required open/closed property of every selected set by the chosen mode.A failure does not stop computation of the union/intersection. Step identifiers and dependenciesRule: closure_each_selected_property · Uses: G001, S002 S004 · impliesBefore mode=closed_intersection;selected member=[0, 1, 2, 3] After required property qualified=Yes Conditions: none Audit the required open/closed property of every selected set by the chosen mode.A failure does not stop computation of the union/intersection. Step identifiers and dependenciesRule: closure_each_selected_property · Uses: G001, S003 S005 · impliesBefore mode=closed_intersection;selected normalized family=[[], [1, 3], [0, 1, 2, 3]] After initial accumulator=[0, 1, 2, 3] Conditions: none 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. Step identifiers and dependenciesRule: empty_union_intersection_identity · Uses: G001, S004 S006 · impliesBefore mode=closed_intersection;accumulator=[0, 1, 2, 3];next member=[] After next accumulator=[] Conditions: none Update using exactly the specified union/intersection operation without substituting a different set or mode. Step identifiers and dependenciesRule: complete_selected_family_set_fold · Uses: G001, S005 S007 · impliesBefore mode=closed_intersection;accumulator=[];next member=[1, 3] After next accumulator=[] Conditions: none Update using exactly the specified union/intersection operation without substituting a different set or mode. Step identifiers and dependenciesRule: complete_selected_family_set_fold · Uses: G001, S006 S008 · impliesBefore mode=closed_intersection;accumulator=[];next member=[0, 1, 2, 3] After next accumulator=[] Conditions: none Update using exactly the specified union/intersection operation without substituting a different set or mode. Step identifiers and dependenciesRule: complete_selected_family_set_fold · Uses: G001, S007 S009 · impliesBefore all selected properties qualified=Yes;result=[] 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. Step identifiers and dependenciesRule: closure_premises_and_result_property_separate · Uses: G001, S008 ## Reverse checks - 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, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: affb09996f00f7bd75387ed7135a96cee04393105276c3554a51678cde29feab - candidate record SHA-256: acde93dec41a278f50206d5eb48fec1d4729b7f74bebc56a189deca8b66bf59d - semantic review receipt SHA-256: bd35da62087a15d6b89b45585dc46e89b9efbe097cba7b3c707b5030496f013c - question-first witness SHA-256: fab45a10d8ac95c7d61d32e47d1bdd3ca862c17c3289a6bf4b01d4dc9492fbb6 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261019-r01-open-closed-set-0090.json)