Closed set

Let $(X,d)$ be a metric space and let $F\subset X$.

The set $F$ is closed if its complement $F^c:=X\setminus F$ is open .

Closed sets are stable under arbitrary intersections and finite unions (see basic properties of closed sets ). The closure of a set is the smallest closed set containing it.

Examples:

• In $\mathbb{R}$, every closed interval $[a,b]$ is closed.
• In any metric space, $\emptyset$ and $X$ are closed.
• In a discrete metric space, every subset of $X$ is closed.