# Definition:Compact Space/Real Analysis/Definition 2

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## Definition

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

Let $H \subseteq \R$.

$H$ 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 $H$.

## Also see

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $5$: Compact spaces: $5.1$: Motivation: Provisional definition $5.1.3$