# 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.