Definition:Closed Set

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

Let $H \subseteq S$.

Then $H$ is closed (in $T\,$) iff its complement $S \setminus H$ is open in $T$.

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

That is, iff $S \setminus H$ is an element of the topology of $T$.

Closed Point
The concept of a closed set can be sharpened to apply to individual points, as follows:

Metric Space
In the context of metric spaces, the same definition applies:

Under Closure Operator
The concept of closure can be made more generally than on a topological space:

Also see

 * Closed Set iff Contains all its Limit Points