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 $\R$ :
 * 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

 * Equivalence of Definitions of Compact Subset of Real Numbers