# Definition:Compact Space/Real Analysis

## Definition

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

Let $H \subseteq \R$.

### Definition 1

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

### Definition 2

$H$ is compact in $\R$ if and only if:

when $H$ is the union of a set of neighborhoods which are open in $H$
$H$ is also the union of a finite number of neighborhoods which are open in $H$.

## Also see

• Results about compact spaces can be found here.