Original question and givens
Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 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, 3, 4], original ordered cover=[[0, 1], [0, 1, 3, 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,valid_subcover?,proves_general_compactness?,one_supplied_cover_does_not_prove_general_compactness).
G001 — Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 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, 3, 4], original ordered cover=[[0, 1], [0, 1, 3, 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,valid_subcover?,proves_general_compactness?,one_supplied_cover_does_not_prove_general_compactness).
Step-by-step derivation
S001 · implies
Before
same complete carrier Fin(5);original ordered complete topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 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 dependencies
Rule: topology_complete_finite_parent_all_pairs · Uses: G001
S002 · implies
Before
original supplied cover index=0;member=[0, 1];same full topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 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 dependencies
Rule: compact_every_original_supplied_member_open · Uses: G001, S001
S003 · implies
Before
original supplied cover index=1;member=[0, 1, 3, 4];same full topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 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 dependencies
Rule: compact_every_original_supplied_member_open · Uses: G001, S002
S004 · implies
Before
original supplied cover index=2;member=[0, 1, 2, 3, 4];same full topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 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 dependencies
Rule: compact_every_original_supplied_member_open · Uses: G001, S003
S005 · implies
Before
same supplied target=[0, 1, 3, 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 dependencies
Rule: compact_supplied_openness_and_coverage_not_universal_theorem · Uses: G001, S004
S006 · implies
Before
original selected indices=[1];sorted duplicate-free indices=[1];original indexed members=[[0, 1, 3, 4]]
After
selected union=[0, 1, 3, 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 dependencies
Rule: compact_selected_original_subcover_whole_family_premise · Uses: G001, S005
S007 · implies
Before
this one supplied cover;selected original indices=[1];valid subcover=Yes
After
proves general compactness from one audit=No;one_supplied_cover_does_not_prove_general_compactness
Conditions: none
The No limits the inference from one audit to a universal property.It does not declare this finite space noncompact;finiteness supplies a separate certificate.Nor does it classify an infinite space from finite tests.
Step identifiers and dependencies
Rule: compact_single_instance_not_all_covers_or_noncompactness · Uses: G001, S006