A compact subset of a Hausdorff space is closed

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Hausdorff space — Distinct points have disjoint open neighborhoods.
• Compact subset — An open cover has a finite subcover.
• Open and closed sets — 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.