Definition:Closed Set

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

Let $H \subseteq X$.

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

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

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

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

Let $A \subseteq B \subseteq X$.

Then $A$ is relatively closed in $B$ iff $A$ is closed in the relative topology of $B$.

Equivalently, $A$ is relatively closed in $B$ iff there is a closed set $C \subseteq X$ with $C \cap B = A$.

This is proved in Relatively Closed by Intersection with Closed Set.

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

Let $a \in X$.

Then $a$ is closed (in $T\,$) iff $\left\{{a}\right\}$ is closed (in $T\,$).

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

Also see

 * Closed Set contains All its Limit Points