Definition:Compact Space/Real Analysis

Definition
Let $\R$ be the real numbers considered as a Euclidean space.

Let $X \subseteq \R$.

Then $X$ is compact in $\R$ iff $X$ is closed and bounded.

Also see

 * Heine–Borel Theorem, where it is proved that this definition is equivalent to the topological definition when $\R$ is considered with the Euclidean topology.