Closed set

A closed set in a topological space $(X,\mathcal{T})$ is a subset $F\subseteq X$ such that its complement $X\setminus F$ (the set difference ) is a open set .

Closed sets are the natural targets of the closure operation: the closure of $A$ is the smallest closed set containing $A$. Closed sets also provide an equivalent formulation of continuity via preimages of closed sets.

Examples:

• In $\mathbb{R}$ with the usual topology, $[0,1]$ is closed.
• In the discrete topology on $X$, every subset of $X$ is closed.
• In the indiscrete topology on $X$, the only closed sets are $\varnothing$ and $X$.