# Question 16542: Closure Membership May Hold outside the Original Subset [Return to catalogue page 83](https://ocme.evemisslab.com/en/questions/page/83/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261101-r02-closure-and-interior-foundations/#q-20261014-advance-20261101-r02-closure-and-interior-0067) 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. R0032-16542 · q-20261014-advance-20261101-r02-closure-and-interior-0067 ## Original question and givens Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=3. Give (point,closure,in_closure?,topology_relative_closure). - G001 — Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Subset=[3, 4], point=3. Give (point,closure,in_closure?,topology_relative_closure). ## 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-closure-and-interior](https://ocme.evemisslab.com/en/mko/mko-closure-and-interior/) ## Step-by-step derivation S001 · impliesBefore same complete carrier Fin(5);original ordered complete topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] After unique empty/whole opens=Yes;ordered open pairs checked=36;first union failure=null;first intersection failure=null Conditions: none Each subset is increasing and duplicate-free.The topology has unique empty/whole opens and closure under every binary union/intersection.Finitely many distinct opens reduce arbitrary unions to finite ones,handled by binary closure and the empty set.No metric is supplied.Keep original topology order. Step identifiers and dependenciesRule: topology_complete_finite_parent_all_pairs · Uses: G001 S002 · impliesBefore selected subset:original open index=0;open=[];same-carrier closed complement=[0, 1, 2, 3, 4];subset=[3, 4] After closed set contains whole subset=Yes;running intersection of qualifying closed sets=[0, 1, 2, 3, 4] Conditions: none Complement every original open in the same carrier to obtain all closed sets.Keep exactly those containing all of A,and intersect every qualifying set rather than one convenient superset.A point outside A can still lie in the final intersection. Step identifiers and dependenciesRule: closure_every_closed_superset_original_open_order · Uses: G001, S001 S003 · impliesBefore selected subset:original open index=1;open=[0];same-carrier closed complement=[1, 2, 3, 4];subset=[3, 4] After closed set contains whole subset=Yes;running intersection of qualifying closed sets=[1, 2, 3, 4] Conditions: none Complement every original open in the same carrier to obtain all closed sets.Keep exactly those containing all of A,and intersect every qualifying set rather than one convenient superset.A point outside A can still lie in the final intersection. Step identifiers and dependenciesRule: closure_every_closed_superset_original_open_order · Uses: G001, S002 S004 · impliesBefore selected subset:original open index=2;open=[0, 3];same-carrier closed complement=[1, 2, 4];subset=[3, 4] After closed set contains whole subset=No;running intersection of qualifying closed sets=[1, 2, 3, 4] Conditions: none Complement every original open in the same carrier to obtain all closed sets.Keep exactly those containing all of A,and intersect every qualifying set rather than one convenient superset.A point outside A can still lie in the final intersection. Step identifiers and dependenciesRule: closure_every_closed_superset_original_open_order · Uses: G001, S003 S005 · impliesBefore selected subset:original open index=3;open=[0, 3, 4];same-carrier closed complement=[1, 2];subset=[3, 4] After closed set contains whole subset=No;running intersection of qualifying closed sets=[1, 2, 3, 4] Conditions: none Complement every original open in the same carrier to obtain all closed sets.Keep exactly those containing all of A,and intersect every qualifying set rather than one convenient superset.A point outside A can still lie in the final intersection. Step identifiers and dependenciesRule: closure_every_closed_superset_original_open_order · Uses: G001, S004 S006 · impliesBefore selected subset:original open index=4;open=[0, 1, 3, 4];same-carrier closed complement=[2];subset=[3, 4] After closed set contains whole subset=No;running intersection of qualifying closed sets=[1, 2, 3, 4] Conditions: none Complement every original open in the same carrier to obtain all closed sets.Keep exactly those containing all of A,and intersect every qualifying set rather than one convenient superset.A point outside A can still lie in the final intersection. Step identifiers and dependenciesRule: closure_every_closed_superset_original_open_order · Uses: G001, S005 S007 · impliesBefore selected subset:original open index=5;open=[0, 1, 2, 3, 4];same-carrier closed complement=[];subset=[3, 4] After closed set contains whole subset=No;running intersection of qualifying closed sets=[1, 2, 3, 4] Conditions: none Complement every original open in the same carrier to obtain all closed sets.Keep exactly those containing all of A,and intersect every qualifying set rather than one convenient superset.A point outside A can still lie in the final intersection. Step identifiers and dependenciesRule: closure_every_closed_superset_original_open_order · Uses: G001, S006 S008 · impliesBefore selected subset:ALL qualifying closed supersets in original complement order=[[0, 1, 2, 3, 4], [1, 2, 3, 4]] After topology-relative closure=[1, 2, 3, 4] Conditions: none The whole carrier is a closed superset,so the family is present.The intersection is the least closed superset in this topology.Original order affects the candidate list,not its set value. Step identifiers and dependenciesRule: closure_all_candidates_intersection_not_metric_guess · Uses: G001, S007 S009 · impliesBefore original point=3;original subset=[3, 4];complete computed set=[1, 2, 3, 4] After membership=Yes;scope=topology_relative_closure Conditions: none Compute the entire closure/interior before testing the original point.Do not substitute original-subset membership or an unspecified distance. Step identifiers and dependenciesRule: topology_point_test_computed_closure_or_interior · 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 (3, [1, 2, 3, 4], Yes, topology_relative_closure) 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: a1d467c20b37273009666361062d7d59017f54595890778baae5d7e0cfeaa227 - candidate record SHA-256: 56c82ac4d346bd696d66c9b769eaeeed7a775e37148aaaa81fff0760db432c9d - semantic review receipt SHA-256: d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e - question-first witness SHA-256: 790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261101-r02-closure-and-interior-0067.json)