# Definition:Compact Space/Euclidean Space

Jump to navigation
Jump to search

## 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 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.

### Complex Analysis

Definition:Compact Space/Complex Analysis

## Also see

- Heine–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): $\text{III}$: Compactness