Dense set

Definition

Let $X$ be a topological space and let $D \subseteq X$. The set $D$ is dense in $X$ when its closure is $X$, that is, $\overline{D} = X$.

Remarks

Equivalently, $D$ is dense if and only if $D$ meets each nonempty open set of $X$. In a metric space, $D$ is dense if and only if for each $x \in X$ and each $\varepsilon > 0$ there is $y \in D$ with $d(x, y) < \varepsilon$. The rationals $\mathbb{Q}$ are dense in $\mathbb{R}$: this is the density of the rationals. The irrationals are also dense in $\mathbb{R}$.

Explanations

L1 human

A dense set has points everywhere in the space: near each point of the space there are points of the dense set. Each real number has rational numbers as close to it as you want.