Definition:Compact Space/Metric Space

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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.