# Definition:Compact Space/Metric Space

## Definition

Let $M = \left({A, d}\right)$ be a metric space.

Let $\tau$ denote the topology on $A$ induced by $d$.

Then $M$ is compact if and only if $\left({A, \tau}\right)$ is a compact topological space.

### Complex Plane

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

• Results about compact spaces can be found here.