# Definition:Compact Space/Real Analysis

< Definition:Compact Space(Redirected from Definition:Compact Subset of Real Numbers)

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## Definition

Let $\R$ be the real number line considered as a topological space under the Euclidean topology.

Let $H \subseteq \R$.

### Definition 1

$H$ is **compact in $\R$** if and only if $H$ is closed and bounded.

### Definition 2

$H$ 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.

## Also see

- 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.