# Question 19068: A Homeomorphism Has Both Forward and Inverse Continuity [Return to catalogue page 96](https://ocme.evemisslab.com/en/questions/page/96/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261105-r05-homeomorphism-foundations/#q-20261014-advance-20261105-r05-homeomorphism-0018) 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. R0037-19068 · q-20261014-advance-20261105-r05-homeomorphism-0018 ## Original question and givens Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 3, 4, 2, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate). - G001 — Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 3, 4, 2, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate). ## 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-homeomorphism](https://ocme.evemisslab.com/en/mko/mko-homeomorphism/) ## Step-by-step derivation S001 · impliesBefore source operation=homeo-finite-topology-audit;original snapshot and mathematics retained After explicit ordered source output fields=[source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate] Conditions: none Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore original complete carrier Fin(5);original TOTAL source-indexed candidate map=[0, 3, 4, 2, 1] After all target values distinct and complete;explicit inverse=[0, 4, 3, 1, 2];inverse(map(x))=x and map(inverse(y))=y for EVERY carrier point Conditions: none Accept a bijection only when every target appears exactly once in the complete source list.Build the inverse by original indices and retain both inverse identities rather than trusting a family name. Step identifiers and dependenciesRule: homeo_complete_bijection_not_name_assumption · Uses: G001, S001 S003 · impliesBefore source topology:complete original supplied topology family=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]];carrier Fin(5) After internally sorted subsets without deduplication=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]];canonical whole-family key order retained;empty and FULL carrier present Conditions: none Normalize indices within each subset as the source does,without deduplication or reordering the original family.The family must already have the required key order and no duplicates.This is the complete topology candidate,not a selected good subfamily. Step identifiers and dependenciesRule: homeo_whole_topology_signature_no_missing_opens · Uses: G001, S002 S004 · impliesBefore source topology:left open index=0;left set=[];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[0, 0], [1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [6, 0], [7, 0]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S003 S005 · impliesBefore source topology:left open index=1;left set=[0];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[1, 0], [1, 1], [2, 1], [3, 1], [4, 1], [2, 0], [3, 0], [4, 0]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S004 S006 · impliesBefore source topology:left open index=2;left set=[0, 1, 2];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[2, 0], [2, 1], [2, 2], [3, 2], [3, 1], [2, 5], [3, 5], [3, 0]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S005 S007 · impliesBefore source topology:left open index=3;left set=[0, 1, 2, 3, 4];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[3, 0], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [3, 6], [3, 7]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S006 S008 · impliesBefore source topology:left open index=4;left set=[0, 3, 4];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[4, 0], [4, 1], [3, 1], [3, 4], [4, 4], [3, 0], [3, 7], [4, 7]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S007 S009 · impliesBefore source topology:left open index=5;left set=[1, 2];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[5, 0], [2, 0], [2, 5], [3, 5], [3, 0], [5, 5], [6, 5], [6, 0]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S008 S010 · impliesBefore source topology:left open index=6;left set=[1, 2, 3, 4];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[6, 0], [3, 0], [3, 5], [3, 6], [3, 7], [6, 5], [6, 6], [6, 7]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S009 S011 · impliesBefore source topology:left open index=7;left set=[3, 4];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[7, 0], [4, 0], [3, 0], [3, 7], [4, 7], [6, 0], [6, 7], [7, 7]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S010 S012 · impliesBefore source topology:empty/full carrier and EVERY binary union/intersection checked in COMPLETE finite family of8 opens After ANY arbitrary indexed union has only finitely many DISTINCT opens here,so equals a finite repeated binary union;empty union is empty;finite intersections include empty intersection=full carrier;all topology axioms hold Conditions: none An indexed family may repeat members or have infinitely many indices,but this finite table has only finitely many distinct opens.Removing repetitions preserves its union,so binary closure inductively gives arbitrary-union closure.Finite intersections follow similarly,including the empty cases. Step identifiers and dependenciesRule: homeo_finite_binary_closure_full_topology_quantifiers · Uses: G001, S011 S013 · impliesBefore target topology:complete original supplied topology family=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]];carrier Fin(5) After internally sorted subsets without deduplication=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]];canonical whole-family key order retained;empty and FULL carrier present Conditions: none Normalize indices within each subset as the source does,without deduplication or reordering the original family.The family must already have the required key order and no duplicates.This is the complete topology candidate,not a selected good subfamily. Step identifiers and dependenciesRule: homeo_whole_topology_signature_no_missing_opens · Uses: G001, S012 S014 · impliesBefore target topology:left open index=0;left set=[];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[0, 0], [1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [6, 0], [7, 0]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S013 S015 · impliesBefore target topology:left open index=1;left set=[0];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[1, 0], [1, 1], [2, 1], [3, 1], [4, 1], [2, 0], [3, 0], [4, 0]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S014 S016 · impliesBefore target topology:left open index=2;left set=[0, 1, 2];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[2, 0], [2, 1], [2, 2], [3, 2], [3, 1], [2, 5], [3, 5], [3, 0]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S015 S017 · impliesBefore target topology:left open index=3;left set=[0, 1, 2, 3, 4];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[3, 0], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [3, 6], [3, 7]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S016 S018 · impliesBefore target topology:left open index=4;left set=[0, 3, 4];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[4, 0], [4, 1], [3, 1], [3, 4], [4, 4], [3, 0], [3, 7], [4, 7]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S017 S019 · impliesBefore target topology:left open index=5;left set=[1, 2];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[5, 0], [2, 0], [2, 5], [3, 5], [3, 0], [5, 5], [6, 5], [6, 0]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S018 S020 · impliesBefore target topology:left open index=6;left set=[1, 2, 3, 4];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[6, 0], [3, 0], [3, 5], [3, 6], [3, 7], [6, 5], [6, 6], [6, 7]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S019 S021 · impliesBefore target topology:left open index=7;left set=[3, 4];ALL right open indices in source family order0..7 After ordered [union_open_index,intersection_open_index] for EVERY right open=[[7, 0], [4, 0], [3, 0], [3, 7], [4, 7], [6, 0], [6, 7], [7, 7]] Conditions: none For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure. Step identifiers and dependenciesRule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S020 S022 · impliesBefore target topology:empty/full carrier and EVERY binary union/intersection checked in COMPLETE finite family of8 opens After ANY arbitrary indexed union has only finitely many DISTINCT opens here,so equals a finite repeated binary union;empty union is empty;finite intersections include empty intersection=full carrier;all topology axioms hold Conditions: none An indexed family may repeat members or have infinitely many indices,but this finite table has only finitely many distinct opens.Removing repetitions preserves its union,so binary closure inductively gives arbitrary-union closure.Finite intersections follow similarly,including the empty cases. Step identifiers and dependenciesRule: homeo_finite_binary_closure_full_topology_quantifiers · Uses: G001, S021 S023 · impliesBefore forward map:ACTUAL total map=[0, 3, 4, 2, 1];target open index=0;complete target open=[] After exact full-carrier preimage=[];matching source-open index=0;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S022 S024 · impliesBefore forward map:ACTUAL total map=[0, 3, 4, 2, 1];target open index=1;complete target open=[0] After exact full-carrier preimage=[0];matching source-open index=1;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S023 S025 · impliesBefore forward map:ACTUAL total map=[0, 3, 4, 2, 1];target open index=2;complete target open=[0, 1, 2] After exact full-carrier preimage=[0, 3, 4];matching source-open index=4;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S024 S026 · impliesBefore forward map:ACTUAL total map=[0, 3, 4, 2, 1];target open index=3;complete target open=[0, 1, 2, 3, 4] After exact full-carrier preimage=[0, 1, 2, 3, 4];matching source-open index=3;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S025 S027 · impliesBefore forward map:ACTUAL total map=[0, 3, 4, 2, 1];target open index=4;complete target open=[0, 3, 4] After exact full-carrier preimage=[0, 1, 2];matching source-open index=2;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S026 S028 · impliesBefore forward map:ACTUAL total map=[0, 3, 4, 2, 1];target open index=5;complete target open=[1, 2] After exact full-carrier preimage=[3, 4];matching source-open index=7;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S027 S029 · impliesBefore forward map:ACTUAL total map=[0, 3, 4, 2, 1];target open index=6;complete target open=[1, 2, 3, 4] After exact full-carrier preimage=[1, 2, 3, 4];matching source-open index=6;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S028 S030 · impliesBefore forward map:ACTUAL total map=[0, 3, 4, 2, 1];target open index=7;complete target open=[3, 4] After exact full-carrier preimage=[1, 2];matching source-open index=5;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S029 S031 · impliesBefore forward map:ALL target-open preimage flags=[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes] After continuous=Yes by the COMPLETE open-preimage definition Conditions: none Conjoin every row for continuity in this direction.A failed row is an actual counterexample and is not repaired by bijectivity or continuity in the opposite direction. Step identifiers and dependenciesRule: homeo_full_continuity_each_direction_separate · Uses: G001, S030 S032 · impliesBefore inverse map:ACTUAL total map=[0, 4, 3, 1, 2];target open index=0;complete target open=[] After exact full-carrier preimage=[];matching source-open index=0;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S031 S033 · impliesBefore inverse map:ACTUAL total map=[0, 4, 3, 1, 2];target open index=1;complete target open=[0] After exact full-carrier preimage=[0];matching source-open index=1;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S032 S034 · impliesBefore inverse map:ACTUAL total map=[0, 4, 3, 1, 2];target open index=2;complete target open=[0, 1, 2] After exact full-carrier preimage=[0, 3, 4];matching source-open index=4;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S033 S035 · impliesBefore inverse map:ACTUAL total map=[0, 4, 3, 1, 2];target open index=3;complete target open=[0, 1, 2, 3, 4] After exact full-carrier preimage=[0, 1, 2, 3, 4];matching source-open index=3;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S034 S036 · impliesBefore inverse map:ACTUAL total map=[0, 4, 3, 1, 2];target open index=4;complete target open=[0, 3, 4] After exact full-carrier preimage=[0, 1, 2];matching source-open index=2;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S035 S037 · impliesBefore inverse map:ACTUAL total map=[0, 4, 3, 1, 2];target open index=5;complete target open=[1, 2] After exact full-carrier preimage=[3, 4];matching source-open index=7;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S036 S038 · impliesBefore inverse map:ACTUAL total map=[0, 4, 3, 1, 2];target open index=6;complete target open=[1, 2, 3, 4] After exact full-carrier preimage=[1, 2, 3, 4];matching source-open index=6;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S037 S039 · impliesBefore inverse map:ACTUAL total map=[0, 4, 3, 1, 2];target open index=7;complete target open=[3, 4] After exact full-carrier preimage=[1, 2];matching source-open index=5;preimage open=Yes Conditions: none Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped. Step identifiers and dependenciesRule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S038 S040 · impliesBefore inverse map:ALL target-open preimage flags=[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes] After continuous=Yes by the COMPLETE open-preimage definition Conditions: none Conjoin every row for continuity in this direction.A failed row is an actual counterexample and is not repaired by bijectivity or continuity in the opposite direction. Step identifiers and dependenciesRule: homeo_full_continuity_each_direction_separate · Uses: G001, S039 S041 · impliesBefore complete valid source/target topologies;actual bijection=[0, 3, 4, 2, 1];forward continuity=Yes;inverse continuity=Yes After homeomorphism=Yes is the CONJUNCTION;family name does not assume either continuity direction Conditions: none Valid topologies,bijectivity and the two continuity directions are separate obligations.A continuous bijection still requires inverse continuity.A boundary instance may fail forward continuity or pass both directions;use the actual complete data rather than the family name. Step identifiers and dependenciesRule: homeo_bijection_plus_both_continuities_not_family_assumption · Uses: G001, S040 ## Reverse checks - 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 (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit) 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: 47a7ec2e493bfbf0d50de81697ca9c45c03ef88040402d0d736182b5cbde1559 - candidate record SHA-256: df615d27d7e95f42eefbade30b81a303bb9c587712c4e2d399a7a3c97bde6b72 - semantic review receipt SHA-256: 065507b602ceee60cf47a700d387a51940e0c6724c35a3336a7a7948a199ed80 - question-first witness SHA-256: 7203f33b030792559673b0fee520a8c0e70903c4dd970b35293aa40852f333d6 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261105-r05-homeomorphism-0018.json)