# Compact topological space

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

A space is compact when every open cover has a finite subcover; every finite topological space is compact.

mko-compact-topological-space · definition · v0.10.0



## Mathematical statement

Validate the complete finite carrier and topology, then use the universal finiteness argument: there are finitely many distinct open sets, so every open cover has a finite subcover. Supplied-cover audits are separate; success on one cover does not establish compactness of an arbitrary or infinite space.





## Representative formula (exact TeX)

C(X)





## Assumptions


- The finite carrier and complete topology are explicit.
- General compactness quantifies over every open cover; infinite examples are not classified by finite samples.





## Symbol scope


- C — compact-space certificate
- X — specified topological space





## Intuition, definition, and misconceptions

A finite topology has only finitely many distinct open sets.

An open cover is itself finite as a set family, or an indexed cover can select a finite subcover after deduplicating equal members.

One supplied-cover witness does not establish compactness of an infinite space.

Testing a few covers is not a compactness proof for a general space.





## Adopted derivation guide

[A universal compactness certificate for a finite topology](https://ocme.evemisslab.com/en/derivations/derivation-mko-compact-topological-space-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Topological space](https://ocme.evemisslab.com/en/mko/mko-topological-space/) — The complete open-set family is supplied.
- [Open cover and subcover](https://ocme.evemisslab.com/en/mko/mko-open-cover-and-subcover/) — These are the objects quantified over by compactness.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — Every cover must have some finite subcover.





## Verification boundary

The universal positive certificate is limited to complete finite spaces; named infinite examples require separate certificates.

not_formalized

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





## Next obligation

Add the finite-intersection property and metric 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/compact-topological-space.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-compact-topological-space/index.md)