Use the complete finite topology=[[], [0], [0, 4], [0, 2, 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, 4], [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, 4], [0, 2, 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, 4], [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.
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.
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.
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.
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.
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.
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.
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.