Dense set

Let $(X,d)$ be a metric space and let $D\subseteq X$.

The set $D$ is dense in $X$ if its closure equals $X$: $\overline{D}=X,$ where $\overline{D}$ is the intersection of all closed subsets of $X$ that contain $D$.

Equivalently, $D$ is dense in $X$ if and only if any (hence all) of the following hold:

• For every $x\in X$ and every $\varepsilon>0$, the open ball $B(x,\varepsilon)=\{y\in X:d(x,y)<\varepsilon\}$ intersects $D$: $B(x,\varepsilon)\cap D\neq\varnothing.$
• Every nonempty open set $U\subseteq X$ intersects $D$: $U\neq\varnothing,\ U\text{ open}\ \Rightarrow\ U\cap D\neq\varnothing.$

Dense sets are “topologically large” in the sense that they cannot be avoided by any nontrivial open neighborhood structure.

Examples:

• $\mathbb{Q}$ is dense in $\mathbb{R}$ (usual metric): every interval contains a rational number.
• $\mathbb{R}\setminus\mathbb{Q}$ is dense in $\mathbb{R}$: every interval contains an irrational number.
• $\mathbb{Q}^k$ is dense in $\mathbb{R}^k$.
• $\mathbb{Z}$ is not dense in $\mathbb{R}$ (e.g. $(0,\tfrac12)$ contains no integers).