Definition:Compact Space/Metric Space
< Definition:Compact Space(Redirected from Definition:Compact Metric Space)
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Definition
Let $M = \struct {A, d}$ be a metric space.
Let $\tau$ denote the topology on $A$ induced by $d$.
Then $M$ is compact if and only if $\struct {A, \tau}$ 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.