Step-by-step derivation
S001 · implies
Before
source operation=homeo-finite-topology-audit;original snapshot and mathematics retained
After
explicit ordered source output fields=[source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate]
Conditions: none
Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
original complete carrier Fin(5);original TOTAL source-indexed candidate map=[0, 1, 4, 3, 2]
After
all target values distinct and complete;explicit inverse=[0, 1, 4, 3, 2];inverse(map(x))=x and map(inverse(y))=y for EVERY carrier point
Conditions: none
Accept a bijection only when every target appears exactly once in the complete source list.Build the inverse by original indices and retain both inverse identities rather than trusting a family name.
Step identifiers and dependencies
Rule: homeo_complete_bijection_not_name_assumption · Uses: G001, S001
S003 · implies
Before
source topology:complete original supplied topology family=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]];carrier Fin(5)
After
internally sorted subsets without deduplication=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]];canonical whole-family key order retained;empty and FULL carrier present
Conditions: none
Normalize indices within each subset as the source does,without deduplication or reordering the original family.The family must already have the required key order and no duplicates.This is the complete topology candidate,not a selected good subfamily.
Step identifiers and dependencies
Rule: homeo_whole_topology_signature_no_missing_opens · Uses: G001, S002
S004 · implies
Before
source topology:left open index=0;left set=[];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[0, 0], [1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [6, 0], [7, 0]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S003
S005 · implies
Before
source topology:left open index=1;left set=[0];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[1, 0], [1, 1], [2, 1], [3, 1], [4, 1], [2, 0], [3, 0], [4, 0]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S004
S006 · implies
Before
source topology:left open index=2;left set=[0, 1, 2];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[2, 0], [2, 1], [2, 2], [3, 2], [3, 1], [2, 5], [3, 5], [3, 0]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S005
S007 · implies
Before
source topology:left open index=3;left set=[0, 1, 2, 3, 4];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[3, 0], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [3, 6], [3, 7]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S006
S008 · implies
Before
source topology:left open index=4;left set=[0, 3, 4];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[4, 0], [4, 1], [3, 1], [3, 4], [4, 4], [3, 0], [3, 7], [4, 7]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S007
S009 · implies
Before
source topology:left open index=5;left set=[1, 2];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[5, 0], [2, 0], [2, 5], [3, 5], [3, 0], [5, 5], [6, 5], [6, 0]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S008
S010 · implies
Before
source topology:left open index=6;left set=[1, 2, 3, 4];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[6, 0], [3, 0], [3, 5], [3, 6], [3, 7], [6, 5], [6, 6], [6, 7]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S009
S011 · implies
Before
source topology:left open index=7;left set=[3, 4];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[7, 0], [4, 0], [3, 0], [3, 7], [4, 7], [6, 0], [6, 7], [7, 7]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S010
S012 · implies
Before
source topology:empty/full carrier and EVERY binary union/intersection checked in COMPLETE finite family of8 opens
After
ANY arbitrary indexed union has only finitely many DISTINCT opens here,so equals a finite repeated binary union;empty union is empty;finite intersections include empty intersection=full carrier;all topology axioms hold
Conditions: none
An indexed family may repeat members or have infinitely many indices,but this finite table has only finitely many distinct opens.Removing repetitions preserves its union,so binary closure inductively gives arbitrary-union closure.Finite intersections follow similarly,including the empty cases.
Step identifiers and dependencies
Rule: homeo_finite_binary_closure_full_topology_quantifiers · Uses: G001, S011
S013 · implies
Before
target topology:complete original supplied topology family=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]];carrier Fin(5)
After
internally sorted subsets without deduplication=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]];canonical whole-family key order retained;empty and FULL carrier present
Conditions: none
Normalize indices within each subset as the source does,without deduplication or reordering the original family.The family must already have the required key order and no duplicates.This is the complete topology candidate,not a selected good subfamily.
Step identifiers and dependencies
Rule: homeo_whole_topology_signature_no_missing_opens · Uses: G001, S012
S014 · implies
Before
target topology:left open index=0;left set=[];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[0, 0], [1, 0], [2, 0], [3, 0], [4, 0], [5, 0], [6, 0], [7, 0]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S013
S015 · implies
Before
target topology:left open index=1;left set=[0];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[1, 0], [1, 1], [2, 1], [3, 1], [4, 1], [2, 0], [3, 0], [4, 0]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S014
S016 · implies
Before
target topology:left open index=2;left set=[0, 1, 2, 3, 4];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[2, 0], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [2, 6], [2, 7]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S015
S017 · implies
Before
target topology:left open index=3;left set=[0, 1, 4];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[3, 0], [3, 1], [2, 3], [3, 3], [2, 1], [2, 6], [3, 6], [2, 0]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S016
S018 · implies
Before
target topology:left open index=4;left set=[0, 2, 3];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[4, 0], [4, 1], [2, 4], [2, 1], [4, 4], [2, 7], [2, 0], [4, 7]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S017
S019 · implies
Before
target topology:left open index=5;left set=[1, 2, 3, 4];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[5, 0], [2, 0], [2, 5], [2, 6], [2, 7], [5, 5], [5, 6], [5, 7]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S018
S020 · implies
Before
target topology:left open index=6;left set=[1, 4];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[6, 0], [3, 0], [2, 6], [3, 6], [2, 0], [5, 6], [6, 6], [5, 0]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S019
S021 · implies
Before
target topology:left open index=7;left set=[2, 3];ALL right open indices in source family order0..7
After
ordered [union_open_index,intersection_open_index] for EVERY right open=[[7, 0], [4, 0], [2, 7], [2, 0], [4, 7], [5, 7], [5, 0], [7, 7]]
Conditions: none
For every right open compute the actual union and intersection and locate each in the same complete table.The indices refer to the opens listed above;the entire row is checked rather than sampled closure.
Step identifiers and dependencies
Rule: homeo_every_binary_open_union_intersection_exact_table · Uses: G001, S020
S022 · implies
Before
target topology:empty/full carrier and EVERY binary union/intersection checked in COMPLETE finite family of8 opens
After
ANY arbitrary indexed union has only finitely many DISTINCT opens here,so equals a finite repeated binary union;empty union is empty;finite intersections include empty intersection=full carrier;all topology axioms hold
Conditions: none
An indexed family may repeat members or have infinitely many indices,but this finite table has only finitely many distinct opens.Removing repetitions preserves its union,so binary closure inductively gives arbitrary-union closure.Finite intersections follow similarly,including the empty cases.
Step identifiers and dependencies
Rule: homeo_finite_binary_closure_full_topology_quantifiers · Uses: G001, S021
S023 · implies
Before
forward map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=0;complete target open=[]
After
exact full-carrier preimage=[];matching source-open index=0;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S022
S024 · implies
Before
forward map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=1;complete target open=[0]
After
exact full-carrier preimage=[0];matching source-open index=1;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S023
S025 · implies
Before
forward map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=2;complete target open=[0, 1, 2, 3, 4]
After
exact full-carrier preimage=[0, 1, 2, 3, 4];matching source-open index=3;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S024
S026 · implies
Before
forward map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=3;complete target open=[0, 1, 4]
After
exact full-carrier preimage=[0, 1, 2];matching source-open index=2;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S025
S027 · implies
Before
forward map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=4;complete target open=[0, 2, 3]
After
exact full-carrier preimage=[0, 3, 4];matching source-open index=4;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S026
S028 · implies
Before
forward map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=5;complete target open=[1, 2, 3, 4]
After
exact full-carrier preimage=[1, 2, 3, 4];matching source-open index=6;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S027
S029 · implies
Before
forward map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=6;complete target open=[1, 4]
After
exact full-carrier preimage=[1, 2];matching source-open index=5;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S028
S030 · implies
Before
forward map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=7;complete target open=[2, 3]
After
exact full-carrier preimage=[3, 4];matching source-open index=7;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S029
S031 · implies
Before
forward map:ALL target-open preimage flags=[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]
After
continuous=Yes by the COMPLETE open-preimage definition
Conditions: none
Conjoin every row for continuity in this direction.A failed row is an actual counterexample and is not repaired by bijectivity or continuity in the opposite direction.
Step identifiers and dependencies
Rule: homeo_full_continuity_each_direction_separate · Uses: G001, S030
S032 · implies
Before
inverse map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=0;complete target open=[]
After
exact full-carrier preimage=[];matching source-open index=0;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S031
S033 · implies
Before
inverse map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=1;complete target open=[0]
After
exact full-carrier preimage=[0];matching source-open index=1;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S032
S034 · implies
Before
inverse map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=2;complete target open=[0, 1, 2]
After
exact full-carrier preimage=[0, 1, 4];matching source-open index=3;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S033
S035 · implies
Before
inverse map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=3;complete target open=[0, 1, 2, 3, 4]
After
exact full-carrier preimage=[0, 1, 2, 3, 4];matching source-open index=2;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S034
S036 · implies
Before
inverse map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=4;complete target open=[0, 3, 4]
After
exact full-carrier preimage=[0, 2, 3];matching source-open index=4;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S035
S037 · implies
Before
inverse map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=5;complete target open=[1, 2]
After
exact full-carrier preimage=[1, 4];matching source-open index=6;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S036
S038 · implies
Before
inverse map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=6;complete target open=[1, 2, 3, 4]
After
exact full-carrier preimage=[1, 2, 3, 4];matching source-open index=5;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S037
S039 · implies
Before
inverse map:ACTUAL total map=[0, 1, 4, 3, 2];target open index=7;complete target open=[3, 4]
After
exact full-carrier preimage=[2, 3];matching source-open index=7;preimage open=Yes
Conditions: none
Continuity requires the preimage of every target open to be source-open,not an image test or one passing open.None means this preimage is absent from the complete source table,not that computation was skipped.
Step identifiers and dependencies
Rule: homeo_every_target_open_preimage_not_image_or_samples · Uses: G001, S038
S040 · implies
Before
inverse map:ALL target-open preimage flags=[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]
After
continuous=Yes by the COMPLETE open-preimage definition
Conditions: none
Conjoin every row for continuity in this direction.A failed row is an actual counterexample and is not repaired by bijectivity or continuity in the opposite direction.
Step identifiers and dependencies
Rule: homeo_full_continuity_each_direction_separate · Uses: G001, S039
S041 · implies
Before
complete valid source/target topologies;actual bijection=[0, 1, 4, 3, 2];forward continuity=Yes;inverse continuity=Yes
After
homeomorphism=Yes is the CONJUNCTION;family name does not assume either continuity direction
Conditions: none
Valid topologies,bijectivity and the two continuity directions are separate obligations.A continuous bijection still requires inverse continuity.A boundary instance may fail forward continuity or pass both directions;use the actual complete data rather than the family name.
Step identifiers and dependencies
Rule: homeo_bijection_plus_both_continuities_not_family_assumption · Uses: G001, S040