Dense set

A dense set in a topological space $X$ is a subset $D\subseteq X$ such that its closure is all of $X$, i.e. $\overline{D}=X$. Equivalently, $D$ is dense in $X$ if every nonempty open set $U\subseteq X$ intersects $D$.

Density is a way of saying that $D$ is “everywhere in $X$” from the topological viewpoint: every open region contains points of $D$.

Examples:

• In $\mathbb{R}$ with the usual topology, $\mathbb{Q}$ is dense in $\mathbb{R}$.
• In $\mathbb{R}$ with the usual topology, the irrational numbers are dense in $\mathbb{R}$.
• In a discrete space $X$, the only dense subset is $X$ itself.