Definition:Closed Set

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

Let $H \subseteq T$.

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

That is, $H$ is closed iff $T \setminus H \in \vartheta$.

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

Relatively Closed
Let $T$ be a topological space.

Let $A \subseteq B \subseteq T$.

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 $C \subseteq T$ with $C \cap B = A$.

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

Also see

 * Closed Set contains All its Limit Points