Definition:Closed Set/Topology

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Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$.


Definition 1

$H$ is closed (in $T$) if and only if its complement $S \setminus H$ is open in $T$.

That is, $H$ is closed if and only if $\left({S \setminus H}\right) \in \tau$.

That is, if and only if $S \setminus H$ is an element of the topology of $T$.


Definition 2

$H$ is closed (in $T$) if and only if every limit point of $H$ is also a point of $H$.

That is, by the definition of the derived set:

$H$ is closed (in $T$) if and only if $H' \subseteq H$

where $H'$ denotes the derived set of $H$.


Also see

  • Results about closed sets can be found here.