Definition:Compact Space/Metric Space

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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 $\C$


$D$ is bounded in $\C$.

Also see

  • Results about compact spaces can be found here.