# 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 $|X|\leq 2^{2^{d(X)}}$.

 Title dense set Canonical name DenseSet Date of creation 2013-03-22 12:05:42 Last modified on 2013-03-22 12:05:42 Owner yark (2760) Last modified by yark (2760) Numerical id 12 Author yark (2760) Entry type Definition Classification msc 54A99 Synonym dense subset Synonym everywhere dense set Synonym everywhere dense subset Synonym everywhere-dense set Synonym everywhere-dense subset Related topic NowhereDense Related topic DenseInAPoset Defines dense Defines everywhere dense Defines everywhere-dense Defines density