Definition:Compact Space/Real Analysis
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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 $H$.
- Heine–Borel Theorem, where it is proved that this definition is equivalent to the topological definition when $\R$ is considered with the Euclidean topology.
- Results about compact spaces can be found here.