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Question 21229: Normalize All Ambient-Open Intersections to Obtain the Relative Topology

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0041-21229 · q-20261014-advance-20261108-r03-subspace-topology-0004

Original question and givens

Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).

  • G001 — Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], and subset Y=[1, 2, 4]. Its relative topology is exactly every parent-open U intersected with Y, with duplicate intersections and empty intersections normalized. Complements of relative subsets are taken inside Y, not X. Give (subset, induced_topology, open_count, topology_certificate, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source operation=subtop-induced-topology;original snapshot and mathematics retained

After

explicit ordered source output fields=[subset, induced_topology, open_count, topology_certificate, certificate]

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, 3, 4], [0, 1, 3], [0, 1, 3, 4]]

After

ALL [other_open_index,union,intersection] rows=[[0, [], []], [1, [0], []], [2, [0, 1], []], [3, [0, 1, 2, 3, 4], []], [4, [0, 1, 3], []], [5, [0, 1, 3, 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, 3, 4], [0, 1, 3], [0, 1, 3, 4]]

After

ALL [other_open_index,union,intersection] rows=[[0, [0], []], [1, [0], [0]], [2, [0, 1], [0]], [3, [0, 1, 2, 3, 4], [0]], [4, [0, 1, 3], [0]], [5, [0, 1, 3, 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, 3, 4], [0, 1, 3], [0, 1, 3, 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, 3, 4], [0, 1]], [4, [0, 1, 3], [0, 1]], [5, [0, 1, 3, 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, 3, 4];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 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, 3, 4]], [4, [0, 1, 2, 3, 4], [0, 1, 3]], [5, [0, 1, 2, 3, 4], [0, 1, 3, 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, S004

S006 · implies

Before

ORIGINAL parent Fin(5);parent open index=4,U=[0, 1, 3];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]]

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 3], []], [1, [0, 1, 3], [0]], [2, [0, 1, 3], [0, 1]], [3, [0, 1, 2, 3, 4], [0, 1, 3]], [4, [0, 1, 3], [0, 1, 3]], [5, [0, 1, 3, 4], [0, 1, 3]]];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, 3, 4];complete topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]]

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 3, 4], []], [1, [0, 1, 3, 4], [0]], [2, [0, 1, 3, 4], [0, 1]], [3, [0, 1, 2, 3, 4], [0, 1, 3, 4]], [4, [0, 1, 3, 4], [0, 1, 3]], [5, [0, 1, 3, 4], [0, 1, 3, 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, 2, 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, 2, 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, 2, 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, 3, 4];ORIGINAL subset Y=[1, 2, 4]

After

actual U intersect Y=[1, 2, 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, S010

S012 · implies

Before

ORIGINAL parent open index=4,U=[0, 1, 3];ORIGINAL subset Y=[1, 2, 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, S011

S013 · implies

Before

ORIGINAL parent open index=5,U=[0, 1, 3, 4];ORIGINAL subset Y=[1, 2, 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, 2, 4];ALL parent intersections=[[], [], [1], [1, 2, 4], [1], [1, 4]]

After

normalized relative topology=[[], [1], [1, 2, 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

Reverse checks

  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Recompute subtop distinct relative open count with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([1, 2, 4], [[], [1], [1, 2, 4], [1, 4]], 4, Yes, all_parent_open_intersections_normalized)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 40f1729de097ffb84fd58938fa26be9282871d931b5ca67d0114d49ccd085eb0
  • candidate record SHA-256: 3601d1496e5a0042bbdb568d4eda34c6988db27902332ba661d3b74e51600a34
  • semantic review receipt SHA-256: 8f134403487694af4c70d2e00ec84bab542043ef4b3c54517c04f3c409c0262a
  • question-first witness SHA-256: ad772845088ed8e99f87f90d4264b05ea97be35775305a02011af5ee35f0f4ce
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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