Definition:Compact Space/Euclidean Space

Definition
Let $\R^n$ denote Euclidean $n$-space.

Let $H \subseteq \R^n$.

Then $H$ is compact in $\R^n$ $H$ is closed and bounded.

Real Analysis
The same definition applies when $n = 1$, that is, for the real number line:

Let $\R$ be the real number line considered as a topological space under the Euclidean topology.

Let $H \subseteq \R$.

Also see

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