Definition:Compact Space/Real Analysis
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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 $H$ is closed and bounded.
$H$ is compact in $\R$ if and only if:
- when $H$ is the union of a set of neighborhoods which are open in $H$,
- then $H$ is also the union of a finite number of these neighborhoods.
- 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.