Definition:Closed Set

Topology
A subset $$A$$ of a topological space $$X$$ is called closed if its complement $X\setminus A$ is open. (I.e., $$X\setminus A$$ is an element of the topology of $$X$$.)

Equivalent definition
A set $$A\subset X$$ is closed if and only if it contains all of its limit points; see Equivalent Definitions for Closed Set.

Relatively Closed sets
If $$X$$ is a metric space, and $$A\subset B\subset X$$, 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 $$B$$ if and only if there is a closed set $$C\subset X$$ with $$C\cap B = A$$.