# A compact subset of a Hausdorff space is closed

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

If K is compact in the Hausdorff space X, then K is closed in X.

mko-compact-subset-closed-in-hausdorff · theorem · v0.10.0



## Mathematical statement

For each x outside K and each y in K, choose disjoint open U_y containing x and V_y containing y. The V_y cover K, so compactness supplies a finite subcover. The corresponding finite intersection of U_y is an open neighborhood of x disjoint from K; therefore X minus K is open. Empty K is closed too. The Hausdorff hypothesis cannot be removed, and closedness does not conversely imply compactness in general.





## Representative formula (exact TeX)

C(K)=1





## Assumptions


- Exact families use complete finite carriers/topologies and check every Hausdorff pair and relevant open-cover obligation.
- Finite topology certificates prove specified instances only; the general theorem retains both ambient Hausdorffness and compact-subset hypotheses.





## Symbol scope


- C — closedness predicate
- K — compact subset





## Intuition, definition, and misconceptions

Pointwise separation first forms a neighborhood cover of K.

Compactness reduces the possibly infinite cover to finitely many sets, allowing a finite intersection on the x side.

A non-Hausdorff space may have a compact nonclosed subset, and a closed subset need not be compact.

One pair of neighborhoods need not cover all K, and compact subsets are not always closed if the Hausdorff hypothesis is omitted.





## Adopted derivation guide

[Compactness reduces separating neighborhoods to finitely many](https://ocme.evemisslab.com/en/derivations/derivation-mko-compact-subset-closed-in-hausdorff-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Hausdorff space](https://ocme.evemisslab.com/en/mko/mko-hausdorff-space/) — Distinct points have disjoint open neighborhoods.
- [Compact subset](https://ocme.evemisslab.com/en/mko/mko-compact-subset/) — An open cover has a finite subcover.
- [Open and closed sets](https://ocme.evemisslab.com/en/mko/mko-open-closed-set/) — Closedness is equivalent to openness of the complement.





## Verification boundary

Finite model checks do not replace the general proof. Neither the unconditional converse nor the version without Hausdorffness holds.

not_formalized

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





## Next obligation

Add compact Hausdorff spaces and local 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-subset-closed-in-hausdorff.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-compact-subset-closed-in-hausdorff/index.md)