In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, ${\displaystyle A}$ is dense in ${\displaystyle X}$ if the smallest closed subset of ${\displaystyle X}$ containing ${\displaystyle A}$ is ${\displaystyle X}$ itself.

The density of a topological space ${\displaystyle X}$ is the least cardinality of a dense subset of ${\displaystyle X.}$

A subset ${\displaystyle A}$ of a topological space ${\displaystyle X}$ is said to be a dense subset of ${\displaystyle X}$ if any of the following equivalent conditions are satisfied:

and if ${\displaystyle {\mathcal {B}}}$ is a basis of open sets for the topology on ${\displaystyle X}$ then this list can be extended to include:

An alternative definition of dense set in the case of metric spaces is the following. When the topology of ${\displaystyle X}$ is given by a metric, the closure ${\displaystyle {\overline {A}}}$ of ${\displaystyle A}$ in ${\displaystyle X}$ is the union of ${\displaystyle A}$ and the set of all limits of sequences of elements in ${\displaystyle A}$ (its limit points), $${\displaystyle {\overline {A}}=A\cup \left\{\lim _{n\to \infty }a_{n}:a_{n}\in A{\text{ for all }}n\in \mathbb {N} \right\}}$$

Then ${\displaystyle A}$ is dense in ${\displaystyle X}$ if $${\displaystyle {\overline {A}}=X.}$$

If ${\displaystyle \left\{U_{n}\right\}}$ is a sequence of dense open sets in a complete metric space, ${\displaystyle X,}$ then ${\textstyle \bigcap _{n=1}^{\infty }U_{n}}$ is also dense in ${\displaystyle X.}$ This fact is one of the equivalent forms of the Baire category theorem.

The real numbers with the usual topology have the rational numbers as a countable dense subset which shows that the cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The irrational numbers are another dense subset which shows that a topological space may have several disjoint dense subsets (in particular, two dense subsets may be each other's complements), and they need not even be of the same cardinality. Perhaps even more surprisingly, both the rationals and the irrationals have empty interiors, showing that dense sets need not contain any non-empty open set. The intersection of two dense open subsets of a topological space is again dense and open. The empty set is a dense subset of itself. But every dense subset of a non-empty space must also be non-empty.