Definition:Closed Set

Topology
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 metric 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 $$C \cap B = A$$.