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 $\paren {S \setminus H} \in \tau$.
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.