# Definition:Compact Space/Euclidean Space

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

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## Definition

Let $\R^n$ denote Euclidean $n$-space.

Let $H \subseteq \R^n$.

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

### Real Analysis

The same definition applies when $n = 1$, that is, for the real number line:

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.

### Complex Analysis

Let $D$ be a subset of the complex plane $\C$.

Then $D$ is **compact (in $\C$)** if and only if:

- $D$ is closed in $\C$

and

- $D$ is bounded in $\C$.

## Also see

- Heineâ€“Borel Theorem, where it is proved that this definition is equivalent to the topological definition when $\R^n$ 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): Chapter $\text{III}$: Metric Spaces: Compactness