# Topological boundary

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

The boundary of A is closure(A) minus interior(A), equivalently closure(A) intersected with the closure of its complement.

mko-topological-boundary · definition · v0.10.0



## Mathematical statement

After validating the complete finite topology, calculate closure and interior exactly. Retain boundary(A)=boundary(complement A), the empty/whole-set boundaries and dependence on the topology. If a neighborhood-contact criterion is used, every open neighborhood must be considered.





## Representative formula (exact TeX)

B(A)





## Assumptions


- The finite carrier, topology and subset A are complete.
- Complements use the same carrier; no metric boundary is used without a supplied metric.





## Symbol scope


- B — topological boundary
- A — specified subset





## Intuition, definition, and misconceptions

Compute from the complete closure and interior.

A set and its complement have the same boundary.

The empty set and whole space both have empty boundary; the result depends on the topology.

A metric drawing does not determine boundary when the topology has not been specified.





## Adopted derivation guide

[Exact boundary comparison with the complement](https://ocme.evemisslab.com/en/derivations/derivation-mko-topological-boundary-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Topological closure and interior](https://ocme.evemisslab.com/en/mko/mko-closure-and-interior/) — Boundary is closure minus interior.
- [Set operations](https://ocme.evemisslab.com/en/mko/mko-set-operations/) — Difference, complement and intersection define equivalent forms.





## 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 connectedness and separation properties.





## 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/topological-boundary.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-topological-boundary/index.md)