Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.
G001 — Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.
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 and whole opens=Yes;all ordered open pairs=36;first union closure failure=null;first intersection closure failure=null
Conditions: none
Check canonical subsets,empty and whole sets,and union/intersection of every ordered pair.Since the topology family is finite,an arbitrary union involves only finitely many distinct opens and follows by repeated binary unions.Finite intersections follow similarly with the whole set.No metric is supplied or inferred.
original cover family index=0;member=[0, 1];complete same topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]
After
original member open=Yes
Conditions: none
Check openness at every original index.Equal sets at different indices remain distinct indexed members.Do not deduplicate the input or ignore a nonopen member merely because it is unselected.
original cover family index=1;member=[0, 1, 2, 4];complete same topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]
After
original member open=Yes
Conditions: none
Check openness at every original index.Equal sets at different indices remain distinct indexed members.Do not deduplicate the input or ignore a nonopen member merely because it is unselected.
original cover family index=2;member=[0, 1, 2, 3, 4];complete same topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]
After
original member open=Yes
Conditions: none
Check openness at every original index.Equal sets at different indices remain distinct indexed members.Do not deduplicate the input or ignore a nonopen member merely because it is unselected.
same target=[0, 1, 2, 4];union of ALL original members=[0, 1, 2, 3, 4];nonopen original indices=[]
After
covers target=Yes;whole original open cover valid=Yes
Conditions: none
Openness and coverage are separate obligations.The union may exceed the target;containment,not equality,is required.Certify the original cover only when both hold.
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
([], [0, 1, 2, 3, 4], Yes, Yes)
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.