# Definition:Compact Space/Euclidean Space

## Definition

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

Let $H \subseteq \R^n$.

Then $H$ is compact in $\R^n$ if and only if $H$ 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 $H \subseteq \R$.

$H$ is compact in $\R$ if and only if $H$ is closed and bounded.

### Complex Analysis

Let $D$ be a subset of the complex plane $\C$.

Then $D$ is compact (in $\C$) if and only if:

$D$ is closed in $\C$

and

$D$ is bounded in $\C$.

## Also see

• Results about compact spaces can be found here.