# Question 9738: Containment, Openness, and Neighborhood Status Are Different [Return to catalogue page 49](https://ocme.evemisslab.com/en/questions/page/49/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261019-r02-neighborhood-foundations/#q-20261013-advance-20261019-r02-neighborhood-0038) 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. R0018-09738 · q-20261013-advance-20261019-r02-neighborhood-0038 ## Original question and givens Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 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. Point x=0, candidate N=[0, 1, 2, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims. - G001 — Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 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. Point x=0, candidate N=[0, 1, 2, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims. ## 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](https://ocme.evemisslab.com/en/mko/mko-neighborhood/) ## Step-by-step derivation S001 · impliesBefore n=4;raw open family=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]] After universe=[0, 1, 2, 3];normalized opens=[[], [0], [0, 1], [0, 1, 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 point candidate=0;underlying Fin(4) After point qualified=Yes Conditions: none An outside point leaves neighborhood claims unevaluated,not False via negative indexing or an empty witness list.Candidate sets must still independently be typed. Step identifiers and dependenciesRule: neighborhood_point_domain_guard · Uses: G001, S001 S003 · impliesBefore raw candidate=[0, 1, 2, 3] After typed normalized candidate=[0, 1, 2, 3] Conditions: none Set normalization and point qualification are separate data conditions. Step identifiers and dependenciesRule: neighborhood_candidate_set_normalization · Uses: G001, S002 S004 · impliesBefore point=0;complete open family=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]] After all point-containing opens=[[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]] Conditions: none Take every open containing the qualified point as the complete witness candidates.A neighborhood itself need not be open;it must contain at least one of them. Step identifiers and dependenciesRule: complete_point_open_witness_basis · Uses: G001, S003 S005 · impliesBefore phase=candidate;point=0;candidate=[0, 1, 2, 3];point-open basis=[[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]] After all contained open witnesses=[[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]];neighborhood=Yes Conditions: none List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership. Step identifiers and dependenciesRule: all_open_neighborhood_witnesses · Uses: G001, S004 S006 · impliesBefore point qualified=Yes;candidate=[0, 1, 2, 3] After contains point/status=Yes;open/status=Yes;neighborhood/status=Yes Conditions: none Keep point membership,openness and neighborhood status separate.For an outside point the protocol leaves all three fields unevaluated;do not silently fill one from an independent set computation. Step identifiers and dependenciesRule: neighborhood_membership_openness_not_identical · Uses: G001, S005 ## Reverse checks - 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, Yes, [0, 1, 2, 3], Yes, 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: 279a0927301f9f91eb2f73bdec97d1c582a6689b122c80cd4ab1940bf36ab45c - candidate record SHA-256: 88482eea87e0811863f0a522ff830617296ff9302cb7ec75b183f8489f654b1d - 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-r02-neighborhood-0038.json)