# Definition:Compact Space/Real Analysis

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## Definition

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

Let $H \subseteq \R$.

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

## 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.

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{III}$: Compactness - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.9$: Intervals