# Compact subset

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

A subset K of X is compact in its subspace topology exactly when every relative-open cover has a finite subcover.

mko-compact-subset · definition · v0.10.0



## Mathematical statement

A cover must cover K, and a subcover must be selected from the original cover. Passing one cover audit does not prove general compactness. For the complete finite topology and finite subset used here, finiteness supplies compactness, including the empty subset by convention. Do not extrapolate this finite enumeration to infinite spaces or identify compactness with ambient closedness and boundedness without extra hypotheses.





## Representative formula (exact TeX)

K\subseteq X





## Assumptions


- The complete finite ambient topology, subset K and induced topology are explicit.
- Cover members are relative-open sets from the complete induced topology.





## Symbol scope


- K — compact subset
- X — ambient topological space





## Intuition, definition, and misconceptions

Compactness is relative to the subspace topology on K.

A finite subcover is selected from the original cover.

The finite-space theorem cannot be extrapolated by finite sampling to an infinite topology.

Checking one cover does not prove compactness, and compactness is not universally closed-and-bounded.





## Adopted derivation guide

[Compactness of a finite subspace](https://ocme.evemisslab.com/en/derivations/derivation-mko-compact-subset-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Compact topological space](https://ocme.evemisslab.com/en/mko/mko-compact-topological-space/) — Compactness uses the open-cover definition.
- [Subspace topology](https://ocme.evemisslab.com/en/mko/mko-subspace-topology/) — The relative topology is fixed.





## Verification boundary

The general positive certificate in this slice is restricted to complete finite spaces.

not_formalized

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





## Next obligation

Add continuous images and closedness under Hausdorff hypotheses.





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