# Definition:Compact Space/Metric Space

< Definition:Compact Space(Redirected from Definition:Compact Metric Space)

Jump to navigation
Jump to search
## 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.