Original question and givens
Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 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. Subset=[1, 3], point=4. Give (point,closure,in_closure?,topology_relative_closure).
G001 — Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 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. Subset=[1, 3], point=4. Give (point,closure,in_closure?,topology_relative_closure).
Step-by-step derivation
S001 · implies
Before
same complete carrier Fin(5);original ordered complete topology=[[], [0], [0, 3], [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
selected subset:original open index=0;open=[];same-carrier closed complement=[0, 1, 2, 3, 4];subset=[1, 3]
After
closed set contains whole subset=Yes;running intersection of qualifying closed sets=[0, 1, 2, 3, 4]
Conditions: none
Complement every original open in the same carrier to obtain all closed sets.Keep exactly those containing all of A,and intersect every qualifying set rather than one convenient superset.A point outside A can still lie in the final intersection.
Step identifiers and dependencies
Rule: closure_every_closed_superset_original_open_order · Uses: G001, S001
S003 · implies
Before
selected subset:original open index=1;open=[0];same-carrier closed complement=[1, 2, 3, 4];subset=[1, 3]
After
closed set contains whole subset=Yes;running intersection of qualifying closed sets=[1, 2, 3, 4]
Conditions: none
Complement every original open in the same carrier to obtain all closed sets.Keep exactly those containing all of A,and intersect every qualifying set rather than one convenient superset.A point outside A can still lie in the final intersection.
Step identifiers and dependencies
Rule: closure_every_closed_superset_original_open_order · Uses: G001, S002
S004 · implies
Before
selected subset:original open index=2;open=[0, 3];same-carrier closed complement=[1, 2, 4];subset=[1, 3]
After
closed set contains whole subset=No;running intersection of qualifying closed sets=[1, 2, 3, 4]
Conditions: none
Complement every original open in the same carrier to obtain all closed sets.Keep exactly those containing all of A,and intersect every qualifying set rather than one convenient superset.A point outside A can still lie in the final intersection.
Step identifiers and dependencies
Rule: closure_every_closed_superset_original_open_order · Uses: G001, S003
S005 · implies
Before
selected subset:original open index=3;open=[0, 1, 3];same-carrier closed complement=[2, 4];subset=[1, 3]
After
closed set contains whole subset=No;running intersection of qualifying closed sets=[1, 2, 3, 4]
Conditions: none
Complement every original open in the same carrier to obtain all closed sets.Keep exactly those containing all of A,and intersect every qualifying set rather than one convenient superset.A point outside A can still lie in the final intersection.
Step identifiers and dependencies
Rule: closure_every_closed_superset_original_open_order · Uses: G001, S004
S006 · implies
Before
selected subset:original open index=4;open=[0, 1, 3, 4];same-carrier closed complement=[2];subset=[1, 3]
After
closed set contains whole subset=No;running intersection of qualifying closed sets=[1, 2, 3, 4]
Conditions: none
Complement every original open in the same carrier to obtain all closed sets.Keep exactly those containing all of A,and intersect every qualifying set rather than one convenient superset.A point outside A can still lie in the final intersection.
Step identifiers and dependencies
Rule: closure_every_closed_superset_original_open_order · Uses: G001, S005
S007 · implies
Before
selected subset:original open index=5;open=[0, 1, 2, 3, 4];same-carrier closed complement=[];subset=[1, 3]
After
closed set contains whole subset=No;running intersection of qualifying closed sets=[1, 2, 3, 4]
Conditions: none
Complement every original open in the same carrier to obtain all closed sets.Keep exactly those containing all of A,and intersect every qualifying set rather than one convenient superset.A point outside A can still lie in the final intersection.
Step identifiers and dependencies
Rule: closure_every_closed_superset_original_open_order · Uses: G001, S006
S008 · implies
Before
selected subset:ALL qualifying closed supersets in original complement order=[[0, 1, 2, 3, 4], [1, 2, 3, 4]]
After
topology-relative closure=[1, 2, 3, 4]
Conditions: none
The whole carrier is a closed superset,so the family is present.The intersection is the least closed superset in this topology.Original order affects the candidate list,not its set value.
Step identifiers and dependencies
Rule: closure_all_candidates_intersection_not_metric_guess · Uses: G001, S007
S009 · implies
Before
original point=4;original subset=[1, 3];complete computed set=[1, 2, 3, 4]
After
membership=Yes;scope=topology_relative_closure
Conditions: none
Compute the entire closure/interior before testing the original point.Do not substitute original-subset membership or an unspecified distance.
Step identifiers and dependencies
Rule: topology_point_test_computed_closure_or_interior · Uses: G001, S008