Dense subset

Let $(X,d)$ be a metric space and let $D\subseteq X$. The set $D$ is dense in $X$ if

(see closure ). Equivalently, $D$ is dense in $X$ iff for every nonempty open set $U\subseteq X$, one has $U\cap D\neq\varnothing$.

Density means that every point of $X$ can be approximated arbitrarily well by points of $D$. Dense subsets are central in approximation theorems and in separability questions.

Examples:

• $\mathbb{Q}$ is dense in $\mathbb{R}$.
• The set $\{(q_1,\dots,q_k): q_i\in\mathbb{Q}\}$ is dense in $\mathbb{R}^k$.
• $\mathbb{Z}$ is not dense in $\mathbb{R}$ (e.g., $(0,1)$ contains no integers).