Definition:Closed Set

Definition
A subset $A$ of a topological space $X$ is called closed if its complement $X \setminus A$ is open.

That is, $A$ is closed iff $X \setminus A$ is an element of the topology of $X$.

Equivalent definition
A set $A \subseteq X$ is closed iff it contains all of its limit points.

See Equivalent Definitions for Closed Set.

Relatively Closed sets
If $X$ is a topological space, and $A \subseteq B \subseteq X$, then we say that $A$ is relatively closed in $B$ if $A$ 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 $C \cap B = A$.