# Question 9226: Topology Qualification before Discrete or Indiscrete Classification [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-0051) 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. R0017-09226 · q-20261013-advance-20261018-r01-topological-space-0051 ## Original question and givens Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide. - G001 — Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide. ## 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=given;n=4;raw family=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]] After universe=[0, 1, 2, 3];canonical opens=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 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 · impliesBefore space=given;canonical family=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]] After self pairs are idempotent;unordered distinct pairs cover all ordered pairs Conditions: none Self pairs are covered by A∪A=A and A∩A=A.Commutativity means all distinct unordered pairs cover every ordered pair;this is not sampling. Step identifiers and dependenciesRule: binary_set_operation_symmetry_coverage · Uses: G001, S001 S003 · impliesBefore space=given;A=[];B=[0] After union=[0];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S002 S004 · impliesBefore space=given;A=[];B=[1] After union=[1];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S003 S005 · impliesBefore space=given;A=[];B=[0, 1] After union=[0, 1];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S004 S006 · impliesBefore space=given;A=[];B=[2] After union=[2];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S005 S007 · impliesBefore space=given;A=[];B=[0, 2] After union=[0, 2];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S006 S008 · impliesBefore space=given;A=[];B=[1, 2] After union=[1, 2];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S007 S009 · impliesBefore space=given;A=[];B=[0, 1, 2] After union=[0, 1, 2];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S008 S010 · impliesBefore space=given;A=[];B=[3] After union=[3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S009 S011 · impliesBefore space=given;A=[];B=[0, 3] After union=[0, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S010 S012 · impliesBefore space=given;A=[];B=[1, 3] After union=[1, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S011 S013 · impliesBefore space=given;A=[];B=[0, 1, 3] After union=[0, 1, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S012 S014 · impliesBefore space=given;A=[];B=[2, 3] After union=[2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S013 S015 · impliesBefore space=given;A=[];B=[0, 2, 3] After union=[0, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S014 S016 · impliesBefore space=given;A=[];B=[1, 2, 3] After union=[1, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S015 S017 · impliesBefore space=given;A=[];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S016 S018 · impliesBefore space=given;A=[0];B=[1] After union=[0, 1];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S017 S019 · impliesBefore space=given;A=[0];B=[0, 1] After union=[0, 1];in family=Yes;intersection=[0];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S018 S020 · impliesBefore space=given;A=[0];B=[2] After union=[0, 2];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S019 S021 · impliesBefore space=given;A=[0];B=[0, 2] After union=[0, 2];in family=Yes;intersection=[0];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S020 S022 · impliesBefore space=given;A=[0];B=[1, 2] After union=[0, 1, 2];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S021 S023 · impliesBefore space=given;A=[0];B=[0, 1, 2] After union=[0, 1, 2];in family=Yes;intersection=[0];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S022 S024 · impliesBefore space=given;A=[0];B=[3] After union=[0, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S023 S025 · impliesBefore space=given;A=[0];B=[0, 3] After union=[0, 3];in family=Yes;intersection=[0];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S024 S026 · impliesBefore space=given;A=[0];B=[1, 3] After union=[0, 1, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S025 S027 · impliesBefore space=given;A=[0];B=[0, 1, 3] After union=[0, 1, 3];in family=Yes;intersection=[0];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S026 S028 · impliesBefore space=given;A=[0];B=[2, 3] After union=[0, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S027 S029 · impliesBefore space=given;A=[0];B=[0, 2, 3] After union=[0, 2, 3];in family=Yes;intersection=[0];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S028 S030 · impliesBefore space=given;A=[0];B=[1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S029 S031 · impliesBefore space=given;A=[0];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[0];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S030 S032 · impliesBefore space=given;A=[1];B=[0, 1] After union=[0, 1];in family=Yes;intersection=[1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S031 S033 · impliesBefore space=given;A=[1];B=[2] After union=[1, 2];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S032 S034 · impliesBefore space=given;A=[1];B=[0, 2] After union=[0, 1, 2];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S033 S035 · impliesBefore space=given;A=[1];B=[1, 2] After union=[1, 2];in family=Yes;intersection=[1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S034 S036 · impliesBefore space=given;A=[1];B=[0, 1, 2] After union=[0, 1, 2];in family=Yes;intersection=[1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S035 S037 · impliesBefore space=given;A=[1];B=[3] After union=[1, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S036 S038 · impliesBefore space=given;A=[1];B=[0, 3] After union=[0, 1, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S037 S039 · impliesBefore space=given;A=[1];B=[1, 3] After union=[1, 3];in family=Yes;intersection=[1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S038 S040 · impliesBefore space=given;A=[1];B=[0, 1, 3] After union=[0, 1, 3];in family=Yes;intersection=[1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S039 S041 · impliesBefore space=given;A=[1];B=[2, 3] After union=[1, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S040 S042 · impliesBefore space=given;A=[1];B=[0, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S041 S043 · impliesBefore space=given;A=[1];B=[1, 2, 3] After union=[1, 2, 3];in family=Yes;intersection=[1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S042 S044 · impliesBefore space=given;A=[1];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S043 S045 · impliesBefore space=given;A=[0, 1];B=[2] After union=[0, 1, 2];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S044 S046 · impliesBefore space=given;A=[0, 1];B=[0, 2] After union=[0, 1, 2];in family=Yes;intersection=[0];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S045 S047 · impliesBefore space=given;A=[0, 1];B=[1, 2] After union=[0, 1, 2];in family=Yes;intersection=[1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S046 S048 · impliesBefore space=given;A=[0, 1];B=[0, 1, 2] After union=[0, 1, 2];in family=Yes;intersection=[0, 1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S047 S049 · impliesBefore space=given;A=[0, 1];B=[3] After union=[0, 1, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S048 S050 · impliesBefore space=given;A=[0, 1];B=[0, 3] After union=[0, 1, 3];in family=Yes;intersection=[0];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S049 S051 · impliesBefore space=given;A=[0, 1];B=[1, 3] After union=[0, 1, 3];in family=Yes;intersection=[1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S050 S052 · impliesBefore space=given;A=[0, 1];B=[0, 1, 3] After union=[0, 1, 3];in family=Yes;intersection=[0, 1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S051 S053 · impliesBefore space=given;A=[0, 1];B=[2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S052 S054 · impliesBefore space=given;A=[0, 1];B=[0, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[0];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S053 S055 · impliesBefore space=given;A=[0, 1];B=[1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S054 S056 · impliesBefore space=given;A=[0, 1];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[0, 1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S055 S057 · impliesBefore space=given;A=[2];B=[0, 2] After union=[0, 2];in family=Yes;intersection=[2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S056 S058 · impliesBefore space=given;A=[2];B=[1, 2] After union=[1, 2];in family=Yes;intersection=[2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S057 S059 · impliesBefore space=given;A=[2];B=[0, 1, 2] After union=[0, 1, 2];in family=Yes;intersection=[2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S058 S060 · impliesBefore space=given;A=[2];B=[3] After union=[2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S059 S061 · impliesBefore space=given;A=[2];B=[0, 3] After union=[0, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S060 S062 · impliesBefore space=given;A=[2];B=[1, 3] After union=[1, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S061 S063 · impliesBefore space=given;A=[2];B=[0, 1, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S062 S064 · impliesBefore space=given;A=[2];B=[2, 3] After union=[2, 3];in family=Yes;intersection=[2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S063 S065 · impliesBefore space=given;A=[2];B=[0, 2, 3] After union=[0, 2, 3];in family=Yes;intersection=[2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S064 S066 · impliesBefore space=given;A=[2];B=[1, 2, 3] After union=[1, 2, 3];in family=Yes;intersection=[2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S065 S067 · impliesBefore space=given;A=[2];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S066 S068 · impliesBefore space=given;A=[0, 2];B=[1, 2] After union=[0, 1, 2];in family=Yes;intersection=[2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S067 S069 · impliesBefore space=given;A=[0, 2];B=[0, 1, 2] After union=[0, 1, 2];in family=Yes;intersection=[0, 2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S068 S070 · impliesBefore space=given;A=[0, 2];B=[3] After union=[0, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S069 S071 · impliesBefore space=given;A=[0, 2];B=[0, 3] After union=[0, 2, 3];in family=Yes;intersection=[0];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S070 S072 · impliesBefore space=given;A=[0, 2];B=[1, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S071 S073 · impliesBefore space=given;A=[0, 2];B=[0, 1, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[0];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S072 S074 · impliesBefore space=given;A=[0, 2];B=[2, 3] After union=[0, 2, 3];in family=Yes;intersection=[2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S073 S075 · impliesBefore space=given;A=[0, 2];B=[0, 2, 3] After union=[0, 2, 3];in family=Yes;intersection=[0, 2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S074 S076 · impliesBefore space=given;A=[0, 2];B=[1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S075 S077 · impliesBefore space=given;A=[0, 2];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[0, 2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S076 S078 · impliesBefore space=given;A=[1, 2];B=[0, 1, 2] After union=[0, 1, 2];in family=Yes;intersection=[1, 2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S077 S079 · impliesBefore space=given;A=[1, 2];B=[3] After union=[1, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S078 S080 · impliesBefore space=given;A=[1, 2];B=[0, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S079 S081 · impliesBefore space=given;A=[1, 2];B=[1, 3] After union=[1, 2, 3];in family=Yes;intersection=[1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S080 S082 · impliesBefore space=given;A=[1, 2];B=[0, 1, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S081 S083 · impliesBefore space=given;A=[1, 2];B=[2, 3] After union=[1, 2, 3];in family=Yes;intersection=[2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S082 S084 · impliesBefore space=given;A=[1, 2];B=[0, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S083 S085 · impliesBefore space=given;A=[1, 2];B=[1, 2, 3] After union=[1, 2, 3];in family=Yes;intersection=[1, 2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S084 S086 · impliesBefore space=given;A=[1, 2];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[1, 2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S085 S087 · impliesBefore space=given;A=[0, 1, 2];B=[3] After union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S086 S088 · impliesBefore space=given;A=[0, 1, 2];B=[0, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[0];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S087 S089 · impliesBefore space=given;A=[0, 1, 2];B=[1, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S088 S090 · impliesBefore space=given;A=[0, 1, 2];B=[0, 1, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[0, 1];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S089 S091 · impliesBefore space=given;A=[0, 1, 2];B=[2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S090 S092 · impliesBefore space=given;A=[0, 1, 2];B=[0, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[0, 2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S091 S093 · impliesBefore space=given;A=[0, 1, 2];B=[1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[1, 2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S092 S094 · impliesBefore space=given;A=[0, 1, 2];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[0, 1, 2];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S093 S095 · impliesBefore space=given;A=[3];B=[0, 3] After union=[0, 3];in family=Yes;intersection=[3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S094 S096 · impliesBefore space=given;A=[3];B=[1, 3] After union=[1, 3];in family=Yes;intersection=[3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S095 S097 · impliesBefore space=given;A=[3];B=[0, 1, 3] After union=[0, 1, 3];in family=Yes;intersection=[3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S096 S098 · impliesBefore space=given;A=[3];B=[2, 3] After union=[2, 3];in family=Yes;intersection=[3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S097 S099 · impliesBefore space=given;A=[3];B=[0, 2, 3] After union=[0, 2, 3];in family=Yes;intersection=[3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S098 S100 · impliesBefore space=given;A=[3];B=[1, 2, 3] After union=[1, 2, 3];in family=Yes;intersection=[3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S099 S101 · impliesBefore space=given;A=[3];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S100 S102 · impliesBefore space=given;A=[0, 3];B=[1, 3] After union=[0, 1, 3];in family=Yes;intersection=[3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S101 S103 · impliesBefore space=given;A=[0, 3];B=[0, 1, 3] After union=[0, 1, 3];in family=Yes;intersection=[0, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S102 S104 · impliesBefore space=given;A=[0, 3];B=[2, 3] After union=[0, 2, 3];in family=Yes;intersection=[3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S103 S105 · impliesBefore space=given;A=[0, 3];B=[0, 2, 3] After union=[0, 2, 3];in family=Yes;intersection=[0, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S104 S106 · impliesBefore space=given;A=[0, 3];B=[1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S105 S107 · impliesBefore space=given;A=[0, 3];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[0, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S106 S108 · impliesBefore space=given;A=[1, 3];B=[0, 1, 3] After union=[0, 1, 3];in family=Yes;intersection=[1, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S107 S109 · impliesBefore space=given;A=[1, 3];B=[2, 3] After union=[1, 2, 3];in family=Yes;intersection=[3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S108 S110 · impliesBefore space=given;A=[1, 3];B=[0, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S109 S111 · impliesBefore space=given;A=[1, 3];B=[1, 2, 3] After union=[1, 2, 3];in family=Yes;intersection=[1, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S110 S112 · impliesBefore space=given;A=[1, 3];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[1, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S111 S113 · impliesBefore space=given;A=[0, 1, 3];B=[2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S112 S114 · impliesBefore space=given;A=[0, 1, 3];B=[0, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[0, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S113 S115 · impliesBefore space=given;A=[0, 1, 3];B=[1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[1, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S114 S116 · impliesBefore space=given;A=[0, 1, 3];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[0, 1, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S115 S117 · impliesBefore space=given;A=[2, 3];B=[0, 2, 3] After union=[0, 2, 3];in family=Yes;intersection=[2, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S116 S118 · impliesBefore space=given;A=[2, 3];B=[1, 2, 3] After union=[1, 2, 3];in family=Yes;intersection=[2, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S117 S119 · impliesBefore space=given;A=[2, 3];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[2, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S118 S120 · impliesBefore space=given;A=[0, 2, 3];B=[1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[2, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S119 S121 · impliesBefore space=given;A=[0, 2, 3];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[0, 2, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S120 S122 · impliesBefore space=given;A=[1, 2, 3];B=[0, 1, 2, 3] After union=[0, 1, 2, 3];in family=Yes;intersection=[1, 2, 3];in family=Yes Conditions: none Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure. Step identifiers and dependenciesRule: finite_topology_complete_binary_pair · Uses: G001, S121 S123 · impliesBefore space=given;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, S122 S124 · impliesBefore topology qualified=Yes;family size=16;powerset size=16 After classification=discrete Conditions: none Qualify the topology first.All2^n subsets give the discrete topology;otherwise only empty and whole give the indiscrete topology.On the empty space both coincide,and the protocol classifies discrete first. Step identifiers and dependenciesRule: topology_classification_after_qualification · Uses: G001, S123 ## Reverse checks - S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, discrete) 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: 8c04e4134e118df32162c72dac7dc7d4ceb60324a6e8da6c257758d7d349af97 - candidate record SHA-256: 793a93b802f6811ca1df498d6ae989182fc3d522f20da4d96e1986ecee31d399 - 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-0051.json)