Original question and givens
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [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, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [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. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.
G001 — Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [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, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [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. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.
Step-by-step derivation
S001 · implies
Before
same finite carrier Fin(5);complete original ordered topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]
After
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.
Step identifiers and dependencies
Rule: cover_qualified_complete_topology_no_silent_metric · Uses: G001
S002 · implies
Before
original cover family index=0;member=[0, 1];complete same topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [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.
Step identifiers and dependencies
Rule: cover_every_original_index_open_including_unused_members · Uses: G001, S001
S003 · implies
Before
original cover family index=1;member=[0, 1, 2, 3];complete same topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [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.
Step identifiers and dependencies
Rule: cover_every_original_index_open_including_unused_members · Uses: G001, S002
S004 · implies
Before
original cover family index=2;member=[0, 1, 2, 3, 4];complete same topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [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.
Step identifiers and dependencies
Rule: cover_every_original_index_open_including_unused_members · Uses: G001, S003
S005 · implies
Before
same target=[0, 1, 2, 3];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.
Step identifiers and dependencies
Rule: cover_openness_and_target_coverage_separate_conjunction · Uses: G001, S004
S006 · implies
Before
raw selected ORIGINAL indices=[1];normalized duplicate-free indices=[1];selected original sets=[[0, 1, 2, 3]]
After
selected union=[0, 1, 2, 3];selected all open=Yes;selected covers target=Yes;original family qualified=Yes;valid selected subcover=Yes
Conditions: none
Select only original indexed members,sorting indices but rejecting duplicates.Empty selection can qualify for an empty target.This protocol additionally requires the whole original family to qualify;an otherwise good selected subfamily cannot erase an unselected bad original member.
Step identifiers and dependencies
Rule: cover_selected_original_indices_whole_family_premise_retained · Uses: G001, S005
S007 · implies
Before
remove original selected index=1;remaining selected indices=[];remaining union=[];same target=[0, 1, 2, 3]
After
removal coverage row=[1, [], No]
Conditions: none
Remove this one selected member and recompute coverage of the same target.Keep original indices and removal order.This is inclusion minimality,not a global minimum-cardinality search.
Step identifiers and dependencies
Rule: cover_every_single_member_removal_target_test · Uses: G001, S006
S008 · implies
Before
valid selected subcover=Yes;all single-removal rows=[[1, [], No]]
After
inclusion-minimal=Yes;inclusion_minimal_not_minimum_cardinality
Conditions: none
For a valid subcover,every single deletion must fail.If a proper selected subfamily covered,removing an element outside it would still cover,so these deletion tests suffice for inclusion minimality.An incomparable choice from the original family may use fewer sets;this is not generally minimum cardinality.
Step identifiers and dependencies
Rule: cover_inclusion_minimal_not_global_smallest_size · Uses: G001, S007