# Dense set

 related topics {math, number, function}

In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if any point x in X belongs to A or is a limit point of A.[1]

Formally, a subset A of a topological space X is dense in X if for any point x in X, any neighborhood of x contains at least one point from A. Equivalently, A is dense in X if and only if the only closed subset of X containing A is X itself. This can also be expressed by saying that the closure of A is X, or that the interior of the complement of A is empty.

The density of a topological space X is the least cardinality of a dense subset of X.

## Contents

### Density in metric spaces

An alternative definition of dense set in the case of metric spaces is the following. When the topology of X is given by a metric, the closure $\overline{A}$ of A in X is the union of A and the set of all limits of sequences of elements in A (its limit points),

Then A is dense in X if

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

### Examples

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.

By the Weierstrass approximation theorem, any given complex-valued continuous function defined on a closed interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. In other words, the polynomial functions are dense in the space C[a,b] of continuous complex-valued functions on the interval [a,b], equipped with the supremum norm.