# Topological closure and interior

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

The closure of a set is the intersection of all closed supersets, and its interior is the union of all open subsets, relative to the specified topology.

mko-closure-and-interior · definition · v0.10.0



## Mathematical statement

First validate the complete finite topology, then calculate closure and interior exactly from all closed supersets or open subsets. Retain the empty-set and whole-space boundaries and dependence on the topology; do not silently introduce a metric.





## Representative formula (exact TeX)

C(A)





## Assumptions


- The finite carrier and complete canonical topology family are explicit.
- Every set is a carrier subset and every complement uses that same carrier.





## Symbol scope


- C — closure/interior record
- A — specified subset





## Intuition, definition, and misconceptions

Closure intersects all closed supersets.

Interior unites all open subsets.

The result depends on the topology and is not automatically a metric-ball construction.

One closed superset or one open subset is insufficient, and an unqualified set family must not be treated as a topology.





## Adopted derivation guide

[Closure and interior in a three-point chain topology](https://ocme.evemisslab.com/en/derivations/derivation-mko-closure-and-interior-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Open and closed sets](https://ocme.evemisslab.com/en/mko/mko-open-closed-set/) — Closed supersets and open subsets come from these families.
- [Topological space](https://ocme.evemisslab.com/en/mko/mko-topological-space/) — The complete topology is fixed.
- [Set operations](https://ocme.evemisslab.com/en/mko/mko-set-operations/) — Intersection, union and complement define the constructions.





## Verification boundary

Complete finite enumeration certifies only the specified topology instance.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add boundary, connectedness and compactness.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/closure-and-interior.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-closure-and-interior/index.md)