# Question 16880: A Subcover Witness Is Bound to Its Original Supplied Cover [Return to catalogue page 85](https://ocme.evemisslab.com/en/questions/page/85/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261102-r05-compact-topological-space-foundations/#q-20261014-advance-20261102-r05-compact-topological-space-0055) 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. R0032-16880 · q-20261014-advance-20261102-r05-compact-topological-space-0055 ## Original question and givens Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Target=[0, 1, 2, 4], original ordered cover=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original member must be open and the original union must contain the target. A subcover selects only original family indexes, starting at 0, sorted increasingly with duplicate indexes forbidden. This supplied-cover audit is separate from the universal finite-space compactness theorem. selected=[1]. Give (selected_indexes,covered_union,valid_subcover?,finite_subcover_for_supplied_cover). - G001 — Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Target=[0, 1, 2, 4], original ordered cover=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original member must be open and the original union must contain the target. A subcover selects only original family indexes, starting at 0, sorted increasingly with duplicate indexes forbidden. This supplied-cover audit is separate from the universal finite-space compactness theorem. selected=[1]. Give (selected_indexes,covered_union,valid_subcover?,finite_subcover_for_supplied_cover). ## 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-compact-topological-space](https://ocme.evemisslab.com/en/mko/mko-compact-topological-space/) ## Step-by-step derivation S001 · impliesBefore same complete carrier Fin(5);original ordered complete topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] After unique empty/whole opens=Yes;ordered open pairs checked=36;first union failure=null;first intersection failure=null Conditions: none Each subset is increasing and duplicate-free.The topology has unique empty/whole opens and closure under every binary union/intersection.Finitely many distinct opens reduce arbitrary unions to finite ones,handled by binary closure and the empty set.No metric is supplied.Keep original topology order. Step identifiers and dependenciesRule: topology_complete_finite_parent_all_pairs · Uses: G001 S002 · impliesBefore original supplied cover index=0;member=[0, 1];same full topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] After original member open=Yes Conditions: none Audit every original member's openness,retaining equal sets at distinct indices.An unselected nonopen original member is not erased.This qualifies the supplied family,not the space's compactness. Step identifiers and dependenciesRule: compact_every_original_supplied_member_open · Uses: G001, S001 S003 · impliesBefore original supplied cover index=1;member=[0, 1, 2, 4];same full topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] After original member open=Yes Conditions: none Audit every original member's openness,retaining equal sets at distinct indices.An unselected nonopen original member is not erased.This qualifies the supplied family,not the space's compactness. Step identifiers and dependenciesRule: compact_every_original_supplied_member_open · Uses: G001, S002 S004 · impliesBefore original supplied cover index=2;member=[0, 1, 2, 3, 4];same full topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] After original member open=Yes Conditions: none Audit every original member's openness,retaining equal sets at distinct indices.An unselected nonopen original member is not erased.This qualifies the supplied family,not the space's compactness. Step identifiers and dependenciesRule: compact_every_original_supplied_member_open · Uses: G001, S003 S005 · impliesBefore same supplied target=[0, 1, 2, 4];whole original family union=[0, 1, 2, 3, 4];nonopen indices=[] After covers target=Yes;original supplied cover valid=Yes Conditions: none Coverage means containment of the supplied target;the union may be larger.Both all-member openness and coverage are required.A failed supplied family does not refute the finite-space universal theorem. Step identifiers and dependenciesRule: compact_supplied_openness_and_coverage_not_universal_theorem · Uses: G001, S004 S006 · impliesBefore original selected indices=[1];sorted duplicate-free indices=[1];original indexed members=[[0, 1, 2, 4]] After selected union=[0, 1, 2, 4];selected all open=Yes;selected covers target=Yes;whole original family qualified=Yes;valid subcover=Yes Conditions: none Select only original indexed members,not invented opens.The original family must first qualify;a good selection does not repair a bad unselected original member.Empty selection can cover an empty target. Step identifiers and dependenciesRule: compact_selected_original_subcover_whole_family_premise · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([1], [0, 1, 2, 4], Yes, finite_subcover_for_supplied_cover) 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: 6c4dbead04fa54968f0627762d66e3289c5b08ed0c00334a20eff7745a0d406e - candidate record SHA-256: 7d3d616be1d6c0db4542e293ab479f1921addb42d7cd11fce23415d733b9224e - semantic review receipt SHA-256: d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e - question-first witness SHA-256: 790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261102-r05-compact-topological-space-0055.json)