# Definition:Compact Space/Real Analysis/Definition 2

## Definition

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:

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 $H$.