Step-by-step derivation
S001 · implies
Before
source operation=subtop-ambient-boundary;original snapshot and mathematics retained
After
explicit ordered source output fields=[candidate, relative_open, ambient_open, intersection_certificate, boundary]
Conditions: none
Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL parent Fin(5);parent open index=0,U=[];complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]]
After
ALL [other_open_index,union,intersection] rows=[[0, [], []], [1, [0], []], [2, [0, 1], []], [3, [0, 1, 2], []], [4, [0, 1, 2, 3, 4], []], [5, [0, 1, 2, 4], []]];each belongs to the SAME parent family
Conditions: none
First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.
Step identifiers and dependencies
Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S001
S003 · implies
Before
ORIGINAL parent Fin(5);parent open index=1,U=[0];complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]]
After
ALL [other_open_index,union,intersection] rows=[[0, [0], []], [1, [0], [0]], [2, [0, 1], [0]], [3, [0, 1, 2], [0]], [4, [0, 1, 2, 3, 4], [0]], [5, [0, 1, 2, 4], [0]]];each belongs to the SAME parent family
Conditions: none
First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.
Step identifiers and dependencies
Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S002
S004 · implies
Before
ORIGINAL parent Fin(5);parent open index=2,U=[0, 1];complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]]
After
ALL [other_open_index,union,intersection] rows=[[0, [0, 1], []], [1, [0, 1], [0]], [2, [0, 1], [0, 1]], [3, [0, 1, 2], [0, 1]], [4, [0, 1, 2, 3, 4], [0, 1]], [5, [0, 1, 2, 4], [0, 1]]];each belongs to the SAME parent family
Conditions: none
First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.
Step identifiers and dependencies
Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S003
S005 · implies
Before
ORIGINAL parent Fin(5);parent open index=3,U=[0, 1, 2];complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]]
After
ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2], []], [1, [0, 1, 2], [0]], [2, [0, 1, 2], [0, 1]], [3, [0, 1, 2], [0, 1, 2]], [4, [0, 1, 2, 3, 4], [0, 1, 2]], [5, [0, 1, 2, 4], [0, 1, 2]]];each belongs to the SAME parent family
Conditions: none
First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.
Step identifiers and dependencies
Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S004
S006 · implies
Before
ORIGINAL parent Fin(5);parent open index=4,U=[0, 1, 2, 3, 4];complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]]
After
ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2, 3, 4], []], [1, [0, 1, 2, 3, 4], [0]], [2, [0, 1, 2, 3, 4], [0, 1]], [3, [0, 1, 2, 3, 4], [0, 1, 2]], [4, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4]], [5, [0, 1, 2, 3, 4], [0, 1, 2, 4]]];each belongs to the SAME parent family
Conditions: none
First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.
Step identifiers and dependencies
Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S005
S007 · implies
Before
ORIGINAL parent Fin(5);parent open index=5,U=[0, 1, 2, 4];complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]]
After
ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2, 4], []], [1, [0, 1, 2, 4], [0]], [2, [0, 1, 2, 4], [0, 1]], [3, [0, 1, 2, 4], [0, 1, 2]], [4, [0, 1, 2, 3, 4], [0, 1, 2, 4]], [5, [0, 1, 2, 4], [0, 1, 2, 4]]];each belongs to the SAME parent family
Conditions: none
First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.
Step identifiers and dependencies
Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S006
S008 · implies
Before
ORIGINAL parent open index=0,U=[];ORIGINAL subset Y=[1, 3, 4]
After
actual U intersect Y=[];retain parent index even if another open has the same intersection
Conditions: none
Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.
Step identifiers and dependencies
Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S007
S009 · implies
Before
ORIGINAL parent open index=1,U=[0];ORIGINAL subset Y=[1, 3, 4]
After
actual U intersect Y=[];retain parent index even if another open has the same intersection
Conditions: none
Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.
Step identifiers and dependencies
Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S008
S010 · implies
Before
ORIGINAL parent open index=2,U=[0, 1];ORIGINAL subset Y=[1, 3, 4]
After
actual U intersect Y=[1];retain parent index even if another open has the same intersection
Conditions: none
Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.
Step identifiers and dependencies
Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S009
S011 · implies
Before
ORIGINAL parent open index=3,U=[0, 1, 2];ORIGINAL subset Y=[1, 3, 4]
After
actual U intersect Y=[1];retain parent index even if another open has the same intersection
Conditions: none
Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.
Step identifiers and dependencies
Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S010
S012 · implies
Before
ORIGINAL parent open index=4,U=[0, 1, 2, 3, 4];ORIGINAL subset Y=[1, 3, 4]
After
actual U intersect Y=[1, 3, 4];retain parent index even if another open has the same intersection
Conditions: none
Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.
Step identifiers and dependencies
Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S011
S013 · implies
Before
ORIGINAL parent open index=5,U=[0, 1, 2, 4];ORIGINAL subset Y=[1, 3, 4]
After
actual U intersect Y=[1, 4];retain parent index even if another open has the same intersection
Conditions: none
Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.
Step identifiers and dependencies
Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S012
S014 · calculation
Before
subtop distinct relative open count := sum(1, 1, 1, 1)
After
subtop distinct relative open count = 4
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S013
S015 · implies
Before
ORIGINAL Y=[1, 3, 4];ALL parent intersections=[[], [], [1], [1], [1, 3, 4], [1, 4]]
After
normalized relative topology=[[], [1], [1, 3, 4], [1, 4]];distinct count=4;empty and Y included;unions and finite intersections commute with intersect Y
Conditions: none
Intersection with Y commutes with arbitrary unions and finite intersections;parent closure gives the induced axioms.For empty Y every intersection is empty,so the topology has one open,not zero.
Step identifiers and dependencies
Rule: subtop_induced_axioms_and_empty_Y · Uses: G001, S014
S016 · implies
Before
ORIGINAL candidate=[1, 3, 4] subset Y=[1, 3, 4];ALL original parent opens=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]]
After
ALL ambient U with U intersect Y=candidate=[[0, 1, 2, 3, 4]];relative_open=Yes;ambient_open=No;Y minus candidate=[];relative_closed=Yes
Conditions: none
Relative openness is witnessed by ambient-open intersections,not ambient membership of the candidate.Relative closedness tests the complement inside Y.De Morgan gives the ambient-closed-intersection form;do not substitute X minus candidate for the relative complement.
Step identifiers and dependencies
Rule: subtop_relative_open_witnesses_and_relative_complement · Uses: G001, S015
S017 · implies
Before
ACTUAL candidate=[1, 3, 4];relative_open=Yes;ambient_open=No
After
relative-only certificate=Yes;O intersect Y definition does NOT require O subset Y
Conditions: none
This boundary requires an actually relative-open but non-ambient-open candidate;it does not say every relative open fails ambient openness.
Step identifiers and dependencies
Rule: subtop_boundary_requires_actual_relative_not_ambient · Uses: G001, S016