Question 16585: Interior Membership Needs an Open Neighborhood inside the Subset

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0032-16585 · q-20261014-advance-20261101-r02-closure-and-interior-0085

Original question and givens

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 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=[2, 3], point=1. Give (point,interior,in_interior?,topology_relative_interior).

G001 — Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 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=[2, 3], point=1. Give (point,interior,in_interior?,topology_relative_interior).

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.

mko-closure-and-interior

Step-by-step derivation

S001 · implies

Before

same complete carrier Fin(5);original ordered complete topology=[[], [0], [0, 2], [0, 2, 3], [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

selected subset:original open index=0;open=[];subset=[2, 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, S001

S003 · implies

Before

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

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=[2, 3]

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, 2, 3];subset=[2, 3]

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, 2, 3, 4];subset=[2, 3]

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=[2, 3]

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

S009 · implies

Before

original point=1;original subset=[2, 3];complete computed set=[]

After

membership=No;scope=topology_relative_interior

Conditions: none

Compute the entire closure/interior before testing the original point.Do not substitute original-subset membership or an unspecified distance.

Step identifiers and dependencies

Rule: topology_point_test_computed_closure_or_interior · Uses: G001, S008

Reverse checks

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, [], No, topology_relative_interior)

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: 984a8295c058cca9e5283d9a3de536179d9a5885b4e5675ca907329f2fac35a7
candidate record SHA-256: a69755cf90fecfbf7cd866669140104a640b502404d4096153186cd738f6d921
semantic review receipt SHA-256: d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e
question-first witness SHA-256: 790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8
Lean: NOT_RUN · human review: NOT_PERFORMED

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