# Question 9261: Compare Two Qualified Topologies on the Same Finite Set [Return to catalogue page 47](https://ocme.evemisslab.com/en/questions/page/47/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261018-r01-topological-space-foundations/#q-20261013-advance-20261018-r01-topological-space-0086) 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. R0017-09261 · q-20261013-advance-20261018-r01-topological-space-0086 ## Original question and givens Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes. - G001 — Use X=Fin(4)={0,…,3} and complete actual topologies A=[[], [0, 1, 2, 3]], B=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets by binary-mask order. Give (n, A, B, A⊆B?, B⊆A?, equal/A-coarser/B-coarser/incomparable). Compare families of open sets, not underlying-set sizes. ## 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-topological-space](https://ocme.evemisslab.com/en/mko/mko-topological-space/) ## Step-by-step derivation S001 · impliesBefore space=A;n=4;raw family=[[], [0, 1, 2, 3]] After universe=[0, 1, 2, 3];canonical opens=[[], [0, 1, 2, 3]] Conditions: none Type every candidate as a subset of the same Fin(n),deduplicate elements and members,and order by binary masks.The open family and underlying set are distinct data layers. Step identifiers and dependenciesRule: finite_topology_normalize_relative_universe · Uses: G001 S002 · impliesBefore space=A;empty=Yes;full=Yes;union closed=Yes;intersection closed=Yes After topology=Yes;first union failure=none;first intersection failure=none Conditions: none All four conditions are required.On a finite underlying set there are finitely many distinct opens,so an arbitrary union reduces to a finite distinct-member union,covered by binary closure and the empty union.The empty intersection needs the whole set.Vacuous binary closure does not waive empty/whole-set membership. Step identifiers and dependenciesRule: finite_topology_axioms_joint_gate · Uses: G001, S001 S003 · impliesBefore space=B;n=4;raw family=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]] After universe=[0, 1, 2, 3];canonical opens=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]] Conditions: none Type every candidate as a subset of the same Fin(n),deduplicate elements and members,and order by binary masks.The open family and underlying set are distinct data layers. Step identifiers and dependenciesRule: finite_topology_normalize_relative_universe · Uses: G001, S002 S004 · impliesBefore space=B;empty=Yes;full=Yes;union closed=Yes;intersection closed=Yes After topology=Yes;first union failure=none;first intersection failure=none Conditions: none All four conditions are required.On a finite underlying set there are finitely many distinct opens,so an arbitrary union reduces to a finite distinct-member union,covered by binary closure and the empty union.The empty intersection needs the whole set.Vacuous binary closure does not waive empty/whole-set membership. Step identifiers and dependenciesRule: finite_topology_axioms_joint_gate · Uses: G001, S003 S005 · impliesBefore direction=A in B;open member=[];other family=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]] After member included=Yes Conditions: none Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here. Step identifiers and dependenciesRule: topology_open_family_inclusion_member · Uses: G001, S004 S006 · impliesBefore direction=A in B;open member=[0, 1, 2, 3];other family=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]] After member included=Yes Conditions: none Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here. Step identifiers and dependenciesRule: topology_open_family_inclusion_member · Uses: G001, S005 S007 · impliesBefore direction=B in A;open member=[];other family=[[], [0, 1, 2, 3]] After member included=Yes Conditions: none Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here. Step identifiers and dependenciesRule: topology_open_family_inclusion_member · Uses: G001, S006 S008 · impliesBefore direction=B in A;open member=[2];other family=[[], [0, 1, 2, 3]] After member included=No Conditions: none Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here. Step identifiers and dependenciesRule: topology_open_family_inclusion_member · Uses: G001, S007 S009 · impliesBefore direction=B in A;open member=[2, 3];other family=[[], [0, 1, 2, 3]] After member included=No Conditions: none Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here. Step identifiers and dependenciesRule: topology_open_family_inclusion_member · Uses: G001, S008 S010 · impliesBefore direction=B in A;open member=[0, 2, 3];other family=[[], [0, 1, 2, 3]] After member included=No Conditions: none Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here. Step identifiers and dependenciesRule: topology_open_family_inclusion_member · Uses: G001, S009 S011 · impliesBefore direction=B in A;open member=[0, 1, 2, 3];other family=[[], [0, 1, 2, 3]] After member included=Yes Conditions: none Compare membership of each open set in the other open family,not the underlying-set sizes.The underlying set is fixed here. Step identifiers and dependenciesRule: topology_open_family_inclusion_member · Uses: G001, S010 S012 · impliesBefore A subset B=Yes;B subset A=No After topology relation=A coarser Conditions: none If every A-open is B-open,B is finer and A coarser.Both inclusions mean equality;neither means incomparable. Step identifiers and dependenciesRule: finer_coarser_direction · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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, [[], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, No, A-coarser) 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: 5cb0e24f0f0aa09fe9f9c75a6b648802500eed0d6e52ba4348379b464a5e5b0a - candidate record SHA-256: 316a831d630f943a2a406da35be6a7eedab055f4deae991f2ebb05be4c658e4c - semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8 - question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261018-r01-topological-space-0086.json)