\begin{definition}[Definition:Compact Space/Real Analysis/Definition 2]
Let $\R$ be the real number line considered as a topological space under the Euclidean topology.
Let $H \subseteq \R$.
$H$ is '''compact in $\R$''' {{iff}}:
: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 $H$.
\end{definition}