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Question 16558: Interior Unites All Open Subsets, Not One Convenient Open Set

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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← Question derivation index · Original question batch

R0032-16558 · q-20261014-advance-20261101-r02-closure-and-interior-0033

Original question and givens

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 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, 2]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.

  • G001 — Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 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, 2]. Give (subset,open_subsets,interior), retaining original topology order among the qualifying opens.

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

same complete carrier Fin(5);original ordered complete topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 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=[];subset=[1, 2]

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, S001

S003 · implies

Before

selected subset:original open index=1;open=[0];subset=[1, 2]

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, S002

S004 · implies

Before

selected subset:original open index=2;open=[0, 2];subset=[1, 2]

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, S003

S005 · implies

Before

selected subset:original open index=3;open=[0, 1, 2];subset=[1, 2]

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, S004

S006 · implies

Before

selected subset:original open index=4;open=[0, 1, 2, 4];subset=[1, 2]

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, S005

S007 · implies

Before

selected subset:original open index=5;open=[0, 1, 2, 3, 4];subset=[1, 2]

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, S006

S008 · implies

Before

selected 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, S007

Reverse checks

  • 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], [[]], [])

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: 662bcae11ddd31865d7afd556f8386789091e745812fccb78c9d812590b54647
  • candidate record SHA-256: 97c2fb5e3735cf17ad82c232221f8f92344b280a30997d7e0e28226590c06e06
  • semantic review receipt SHA-256: d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e
  • question-first witness SHA-256: 790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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