Definition:Compact Space/Euclidean Space

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

Let $X \subseteq \R^n$.

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

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

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.