Compact set

Definition

Let $X$ be a topological space and let $K \subseteq X$. The set $K$ is compact when $K$, with the subspace topology, is a compact space.

Remarks

By compactness via open covers, $K$ is compact exactly when each cover of $K$ by open sets of $X$ has a finite subcover. Each finite subset of $X$ is compact. The empty set is compact. In $\mathbb{R}$, the interval $[0, 1]$ is compact, but $(0, 1)$ and $[0, \infty)$ are not. The Heine–Borel theorem describes the compact subsets of $\mathbb{R}^n$.

Explanations

L1 human

A compact set behaves in many ways like a finite set. For example, a continuous real function on a compact set is bounded and attains its maximum, as a function on a finite set does.