# Question 20678: A Cover Alone Does Not Supply the Local Basis Axiom [Return to catalogue page 104](https://ocme.evemisslab.com/en/questions/page/104/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261107-r03-topological-basis-foundations/#q-20261014-advance-20261107-r03-topological-basis-0078) 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. R0040-20678 · q-20261014-advance-20261107-r03-topological-basis-0078 ## Original question and givens Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary). - G001 — Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary). ## 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-basis](https://ocme.evemisslab.com/en/mko/mko-topological-basis/) ## Step-by-step derivation S001 · impliesBefore source operation=topbasis-subbasis-boundary;original snapshot and mathematics retained After explicit ordered source output fields=[covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary] Conditions: none Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore original basis candidate:ORIGINAL complete candidate family=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]];carrier Fin(5) After within-set numeric normalization=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]];no duplicate points or duplicate FAMILY elements;WHOLE original family already in required key order;empty family allowed as candidate Conditions: none A topological basis is a family of subsets,not a vector-space basis;its size may exceed n.Normalize point order within each set without changing original family indices or deleting duplicates to repair data.An empty candidate family can still be audited. Step identifiers and dependenciesRule: topbasis_canonical_candidate_family_not_semantic_reordering · Uses: G001, S001 S003 · impliesBefore COMPLETE original candidate basis=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]];ALL carrier points0..4 After ALL [point,containing_basis_indices] rows=[[0, [0]], [1, [0, 1, 2]], [2, [1, 3]], [3, [2, 3, 4]], [4, [4]]];covers WHOLE carrier=Yes Conditions: none Every carrier point needs a containing candidate element for coverage.This does not replace local intersection refinement.With n≥1 an empty family fails coverage rather than being discarded as an unauditable candidate. Step identifiers and dependenciesRule: topbasis_every_point_cover_separate_from_refinement · Uses: G001, S002 S004 · impliesBefore ORIGINAL ordered basis positions(left,right)=[0, 0];actual basis elements=[[0, 1], [0, 1]];FULL intersection=[0, 1] After ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]], [1, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S003 S005 · impliesBefore ORIGINAL ordered basis positions(left,right)=[0, 1];actual basis elements=[[0, 1], [1, 2]];FULL intersection=[1] After ALL [intersection_point,ALL refining original basis indices] rows=[[1, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S004 S006 · impliesBefore ORIGINAL ordered basis positions(left,right)=[0, 2];actual basis elements=[[0, 1], [1, 3]];FULL intersection=[1] After ALL [intersection_point,ALL refining original basis indices] rows=[[1, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S005 S007 · impliesBefore ORIGINAL ordered basis positions(left,right)=[0, 3];actual basis elements=[[0, 1], [2, 3]];FULL intersection=[] After ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S006 S008 · impliesBefore ORIGINAL ordered basis positions(left,right)=[0, 4];actual basis elements=[[0, 1], [3, 4]];FULL intersection=[] After ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S007 S009 · impliesBefore ORIGINAL ordered basis positions(left,right)=[1, 0];actual basis elements=[[1, 2], [0, 1]];FULL intersection=[1] After ALL [intersection_point,ALL refining original basis indices] rows=[[1, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S008 S010 · impliesBefore ORIGINAL ordered basis positions(left,right)=[1, 1];actual basis elements=[[1, 2], [1, 2]];FULL intersection=[1, 2] After ALL [intersection_point,ALL refining original basis indices] rows=[[1, [1]], [2, [1]]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S009 S011 · impliesBefore ORIGINAL ordered basis positions(left,right)=[1, 2];actual basis elements=[[1, 2], [1, 3]];FULL intersection=[1] After ALL [intersection_point,ALL refining original basis indices] rows=[[1, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S010 S012 · impliesBefore ORIGINAL ordered basis positions(left,right)=[1, 3];actual basis elements=[[1, 2], [2, 3]];FULL intersection=[2] After ALL [intersection_point,ALL refining original basis indices] rows=[[2, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S011 S013 · impliesBefore ORIGINAL ordered basis positions(left,right)=[1, 4];actual basis elements=[[1, 2], [3, 4]];FULL intersection=[] After ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S012 S014 · impliesBefore ORIGINAL ordered basis positions(left,right)=[2, 0];actual basis elements=[[1, 3], [0, 1]];FULL intersection=[1] After ALL [intersection_point,ALL refining original basis indices] rows=[[1, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S013 S015 · impliesBefore ORIGINAL ordered basis positions(left,right)=[2, 1];actual basis elements=[[1, 3], [1, 2]];FULL intersection=[1] After ALL [intersection_point,ALL refining original basis indices] rows=[[1, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S014 S016 · impliesBefore ORIGINAL ordered basis positions(left,right)=[2, 2];actual basis elements=[[1, 3], [1, 3]];FULL intersection=[1, 3] After ALL [intersection_point,ALL refining original basis indices] rows=[[1, [2]], [3, [2]]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S015 S017 · impliesBefore ORIGINAL ordered basis positions(left,right)=[2, 3];actual basis elements=[[1, 3], [2, 3]];FULL intersection=[3] After ALL [intersection_point,ALL refining original basis indices] rows=[[3, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S016 S018 · impliesBefore ORIGINAL ordered basis positions(left,right)=[2, 4];actual basis elements=[[1, 3], [3, 4]];FULL intersection=[3] After ALL [intersection_point,ALL refining original basis indices] rows=[[3, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S017 S019 · impliesBefore ORIGINAL ordered basis positions(left,right)=[3, 0];actual basis elements=[[2, 3], [0, 1]];FULL intersection=[] After ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S018 S020 · impliesBefore ORIGINAL ordered basis positions(left,right)=[3, 1];actual basis elements=[[2, 3], [1, 2]];FULL intersection=[2] After ALL [intersection_point,ALL refining original basis indices] rows=[[2, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S019 S021 · impliesBefore ORIGINAL ordered basis positions(left,right)=[3, 2];actual basis elements=[[2, 3], [1, 3]];FULL intersection=[3] After ALL [intersection_point,ALL refining original basis indices] rows=[[3, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S020 S022 · impliesBefore ORIGINAL ordered basis positions(left,right)=[3, 3];actual basis elements=[[2, 3], [2, 3]];FULL intersection=[2, 3] After ALL [intersection_point,ALL refining original basis indices] rows=[[2, [3]], [3, [3]]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S021 S023 · impliesBefore ORIGINAL ordered basis positions(left,right)=[3, 4];actual basis elements=[[2, 3], [3, 4]];FULL intersection=[3] After ALL [intersection_point,ALL refining original basis indices] rows=[[3, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S022 S024 · impliesBefore ORIGINAL ordered basis positions(left,right)=[4, 0];actual basis elements=[[3, 4], [0, 1]];FULL intersection=[] After ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S023 S025 · impliesBefore ORIGINAL ordered basis positions(left,right)=[4, 1];actual basis elements=[[3, 4], [1, 2]];FULL intersection=[] After ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S024 S026 · impliesBefore ORIGINAL ordered basis positions(left,right)=[4, 2];actual basis elements=[[3, 4], [1, 3]];FULL intersection=[3] After ALL [intersection_point,ALL refining original basis indices] rows=[[3, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S025 S027 · impliesBefore ORIGINAL ordered basis positions(left,right)=[4, 3];actual basis elements=[[3, 4], [2, 3]];FULL intersection=[3] After ALL [intersection_point,ALL refining original basis indices] rows=[[3, []]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S026 S028 · impliesBefore ORIGINAL ordered basis positions(left,right)=[4, 4];actual basis elements=[[3, 4], [3, 4]];FULL intersection=[3, 4] After ALL [intersection_point,ALL refining original basis indices] rows=[[3, [4]], [4, [4]]];empty intersection gives VACUOUS local obligation;first failure retained in source order Conditions: none For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order. Step identifiers and dependenciesRule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S027 S029 · impliesBefore ACTUAL cover=Yes;ALL local point obligations refined=No;first failure[left,right,point,intersection]=[0, 1, 1, [1]] After abstract basis validity=No is cover AND LOCAL refinement;not closure of the family under intersections and not a given-topology certificate yet Conditions: none Their conjunction gives an abstract basis.Coverage alone is insufficient,and a valid basis need not be intersection-closed.A basis for a separately supplied topology also needs openness and generation of every given open. Step identifiers and dependenciesRule: topbasis_cover_and_local_refinement_conjunction · Uses: G001, S028 S030 · impliesBefore ACTUAL complete candidate cover=Yes,local refinement=No,is_basis=No After cover-alone SUFFICES GENERAL CLAIM=No;not a negation of actual is_basis and not a test proving subbasis status false Conditions: none The final No rejects the general cover-only basis inference without negating the independently computed is_basis.A valid basis may still be true.This field does not audit subbasis status,whose usual generation also takes finite intersections before unions. Step identifiers and dependenciesRule: topbasis_cover_alone_No_not_actual_basis_or_subbasis_negation · Uses: G001, S029 ## Reverse checks - 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, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis) 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: 040d31ba063830d38dd8bb7e52a05b55b1de0df1466a9193de347e31470107c7 - candidate record SHA-256: 7579b6a35538825bbc18f6d984f303eea7c83b9e4aecc1fdad5c9ed8f4615482 - semantic review receipt SHA-256: c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d - question-first witness SHA-256: e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261107-r03-topological-basis-0078.json)