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 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
- 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): Chapter $\text{III}$: Metric Spaces: Compactness