dense set

A subset $D$ of a topological space $X$ is said to be dense (or everywhere dense) in $X$ if the closure of $D$ is equal to $X$ . Equivalently, $D$ is dense if and only if $D$ intersects every nonempty open set.

In the special case that $X$ is a metric space with metric $d$ , then this can be rephrased as: for all $\varepsilon > 0$ and all $x\in X$ there is $y\in D$ such that $d(x,y)<\varepsilon$ .

For example, both the rationals $\mathbb{Q}$ and the irrationals $\mathbb{R} \setminus \mathbb{Q}$ are dense in the reals $\mathbb{R}$ .

The least cardinality of a dense set of a topological space is called the density of the space. It is conventional to take the density to be $\aleph_0$ if it would otherwise be finite; with this convention, the spaces of density $\aleph_0$ are precisely the separable spaces. The density of a topological space $X$ is denoted $d(X)$ . If $X$ is a Hausdorff space, it can be shown that $|X| \le 2^{2^{d(X)}}$ .