# Denseness

In mathematics, denseness is an abstract notion that captures the idea that elements of a set A can "approximate" any element of a larger set X, which contains A as a subset, up to arbitrary "accuracy" or "closeness".

## Formal definition

Let X be a topological space. A subset $A \subset X$ is said to be dense in X, or to be a dense set in X, if the closure of A coincides with X (that is, if $\overline{A}=X$); equivalently, the only closed set in X containing A is X itself.

## Examples

1. Consider the set of all rational numbers $\mathbb{Q}$. It can be shown that for an arbitrary real number a and desired accuracy $\epsilon>0$, one can always find some rational number q such that $|q-a|<\epsilon$. Hence the set of rational numbers is dense in the set of real numbers ( $\overline{\mathbb{Q}}=\mathbb{R}$)
2. The set of algebraic polynomials can uniformly approximate any continuous function on a fixed interval [a,b] (with b>a) up to arbitrary accuracy. This is a famous result in analysis known as Weierstrass' theorem. Thus the algebraic polynomials are dense in the space of continuous functions on the interval [a,b] (with respect to the uniform topology). Some content on this page may previously have appeared on Citizendium.