Closure

The closure of a subset $A\subseteq X$ in a topological space $(X,\mathcal{T})$ is

Equivalently, a point $x\in X$ lies in $\overline{A}$ if and only if every neighborhood of $x$ intersects $A$.

A set $F$ is closed exactly when $F=\overline{F}$. Closure is closely tied to limit points and the derived set .

Examples:

• In $\mathbb{R}$ with the usual topology, $\overline{(0,1)}=[0,1]$.
• In $\mathbb{R}$ with the usual topology, $\overline{\mathbb{Q}}=\mathbb{R}$.
• $\overline{\varnothing}=\varnothing$ in any topological space.