Step-by-step derivation
S001 · implies
Before
same complete carrier Fin(5);original ordered complete topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 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
boundary subset:original open index=0;open=[];same-carrier closed complement=[0, 1, 2, 3, 4];subset=[2, 4]
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
boundary subset:original open index=1;open=[0];same-carrier closed complement=[1, 2, 3, 4];subset=[2, 4]
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
boundary subset:original open index=2;open=[0, 4];same-carrier closed complement=[1, 2, 3];subset=[2, 4]
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
boundary subset:original open index=3;open=[0, 2, 4];same-carrier closed complement=[1, 3];subset=[2, 4]
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
boundary subset:original open index=4;open=[0, 2, 3, 4];same-carrier closed complement=[1];subset=[2, 4]
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
boundary subset:original open index=5;open=[0, 1, 2, 3, 4];same-carrier closed complement=[];subset=[2, 4]
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
boundary 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
boundary subset:original open index=0;open=[];subset=[2, 4]
After
whole open contained in subset=Yes;running union of qualifying opens=[]
Conditions: none
The whole open set must lie in A,not merely intersect it.Unite every qualifying open to obtain the largest open subset relative to this topology.
Step identifiers and dependencies
Rule: interior_every_contained_open_original_order · Uses: G001, S008
S010 · implies
Before
boundary subset:original open index=1;open=[0];subset=[2, 4]
After
whole open contained in subset=No;running union of qualifying opens=[]
Conditions: none
The whole open set must lie in A,not merely intersect it.Unite every qualifying open to obtain the largest open subset relative to this topology.
Step identifiers and dependencies
Rule: interior_every_contained_open_original_order · Uses: G001, S009
S011 · implies
Before
boundary subset:original open index=2;open=[0, 4];subset=[2, 4]
After
whole open contained in subset=No;running union of qualifying opens=[]
Conditions: none
The whole open set must lie in A,not merely intersect it.Unite every qualifying open to obtain the largest open subset relative to this topology.
Step identifiers and dependencies
Rule: interior_every_contained_open_original_order · Uses: G001, S010
S012 · implies
Before
boundary subset:original open index=3;open=[0, 2, 4];subset=[2, 4]
After
whole open contained in subset=No;running union of qualifying opens=[]
Conditions: none
The whole open set must lie in A,not merely intersect it.Unite every qualifying open to obtain the largest open subset relative to this topology.
Step identifiers and dependencies
Rule: interior_every_contained_open_original_order · Uses: G001, S011
S013 · implies
Before
boundary subset:original open index=4;open=[0, 2, 3, 4];subset=[2, 4]
After
whole open contained in subset=No;running union of qualifying opens=[]
Conditions: none
The whole open set must lie in A,not merely intersect it.Unite every qualifying open to obtain the largest open subset relative to this topology.
Step identifiers and dependencies
Rule: interior_every_contained_open_original_order · Uses: G001, S012
S014 · implies
Before
boundary subset:original open index=5;open=[0, 1, 2, 3, 4];subset=[2, 4]
After
whole open contained in subset=No;running union of qualifying opens=[]
Conditions: none
The whole open set must lie in A,not merely intersect it.Unite every qualifying open to obtain the largest open subset relative to this topology.
Step identifiers and dependencies
Rule: interior_every_contained_open_original_order · Uses: G001, S013
S015 · implies
Before
boundary subset:ALL contained opens in original topology order=[[]]
After
topology-relative interior=[]
Conditions: none
The empty open always qualifies.The result depends on the specified topology;membership in A alone does not make a point interior.
Step identifiers and dependencies
Rule: interior_all_candidates_union_not_subset_membership · Uses: G001, S014
S016 · implies
Before
boundary subset:same-topology closure=[1, 2, 3, 4];same-topology interior=[]
After
boundary=closure minus interior=[1, 2, 3, 4]
Conditions: none
Subtract the interior from the closure,not from A itself.A point outside A can be a boundary point.No Euclidean drawing or metric is being used.
Step identifiers and dependencies
Rule: boundary_same_topology_closure_minus_interior · Uses: G001, S015
S017 · implies
Before
complement subset:original open index=0;open=[];same-carrier closed complement=[0, 1, 2, 3, 4];subset=[0, 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, S016
S018 · implies
Before
complement subset:original open index=1;open=[0];same-carrier closed complement=[1, 2, 3, 4];subset=[0, 1, 3]
After
closed set contains whole subset=No;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, S017
S019 · implies
Before
complement subset:original open index=2;open=[0, 4];same-carrier closed complement=[1, 2, 3];subset=[0, 1, 3]
After
closed set contains whole subset=No;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, S018
S020 · implies
Before
complement subset:original open index=3;open=[0, 2, 4];same-carrier closed complement=[1, 3];subset=[0, 1, 3]
After
closed set contains whole subset=No;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, S019
S021 · implies
Before
complement subset:original open index=4;open=[0, 2, 3, 4];same-carrier closed complement=[1];subset=[0, 1, 3]
After
closed set contains whole subset=No;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, S020
S022 · implies
Before
complement subset:original open index=5;open=[0, 1, 2, 3, 4];same-carrier closed complement=[];subset=[0, 1, 3]
After
closed set contains whole subset=No;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, S021
S023 · implies
Before
complement subset:ALL qualifying closed supersets in original complement order=[[0, 1, 2, 3, 4]]
After
topology-relative closure=[0, 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, S022
S024 · implies
Before
complement subset:original open index=0;open=[];subset=[0, 1, 3]
After
whole open contained in subset=Yes;running union of qualifying opens=[]
Conditions: none
The whole open set must lie in A,not merely intersect it.Unite every qualifying open to obtain the largest open subset relative to this topology.
Step identifiers and dependencies
Rule: interior_every_contained_open_original_order · Uses: G001, S023
S025 · implies
Before
complement subset:original open index=1;open=[0];subset=[0, 1, 3]
After
whole open contained in subset=Yes;running union of qualifying opens=[0]
Conditions: none
The whole open set must lie in A,not merely intersect it.Unite every qualifying open to obtain the largest open subset relative to this topology.
Step identifiers and dependencies
Rule: interior_every_contained_open_original_order · Uses: G001, S024
S026 · implies
Before
complement subset:original open index=2;open=[0, 4];subset=[0, 1, 3]
After
whole open contained in subset=No;running union of qualifying opens=[0]
Conditions: none
The whole open set must lie in A,not merely intersect it.Unite every qualifying open to obtain the largest open subset relative to this topology.
Step identifiers and dependencies
Rule: interior_every_contained_open_original_order · Uses: G001, S025
S027 · implies
Before
complement subset:original open index=3;open=[0, 2, 4];subset=[0, 1, 3]
After
whole open contained in subset=No;running union of qualifying opens=[0]
Conditions: none
The whole open set must lie in A,not merely intersect it.Unite every qualifying open to obtain the largest open subset relative to this topology.
Step identifiers and dependencies
Rule: interior_every_contained_open_original_order · Uses: G001, S026
S028 · implies
Before
complement subset:original open index=4;open=[0, 2, 3, 4];subset=[0, 1, 3]
After
whole open contained in subset=No;running union of qualifying opens=[0]
Conditions: none
The whole open set must lie in A,not merely intersect it.Unite every qualifying open to obtain the largest open subset relative to this topology.
Step identifiers and dependencies
Rule: interior_every_contained_open_original_order · Uses: G001, S027
S029 · implies
Before
complement subset:original open index=5;open=[0, 1, 2, 3, 4];subset=[0, 1, 3]
After
whole open contained in subset=No;running union of qualifying opens=[0]
Conditions: none
The whole open set must lie in A,not merely intersect it.Unite every qualifying open to obtain the largest open subset relative to this topology.
Step identifiers and dependencies
Rule: interior_every_contained_open_original_order · Uses: G001, S028
S030 · implies
Before
complement subset:ALL contained opens in original topology order=[[], [0]]
After
topology-relative interior=[0]
Conditions: none
The empty open always qualifies.The result depends on the specified topology;membership in A alone does not make a point interior.
Step identifiers and dependencies
Rule: interior_all_candidates_union_not_subset_membership · Uses: G001, S029
S031 · implies
Before
complement subset:same-topology closure=[0, 1, 2, 3, 4];same-topology interior=[0]
After
boundary=closure minus interior=[1, 2, 3, 4]
Conditions: none
Subtract the interior from the closure,not from A itself.A point outside A can be a boundary point.No Euclidean drawing or metric is being used.
Step identifiers and dependencies
Rule: boundary_same_topology_closure_minus_interior · Uses: G001, S030
S032 · implies
Before
same carrier Fin(5);A=[2, 4];complement=[0, 1, 3];computed boundaries=[[1, 2, 3, 4], [1, 2, 3, 4]]
After
interior(A)=X minus closure(X minus A);boundary(A)=closure(A) intersect closure(X minus A);complement swaps the two factors
Conditions: none
In the same carrier,the complement of the interior is the closure of the complement.Thus the boundary is the intersection of the two closures.Complementation swaps the factors,which is the universal same-topology reason.Keep both instantiated computations and do not compare different ambient spaces.
Step identifiers and dependencies
Rule: boundary_complement_duality_same_ambient_topology · Uses: G001, S031