# Denseness

From Citizendium, the Citizens' Compendium

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 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 ); equivalently, the only closed set in *X* containing *A* is *X* itself.

## Examples

- Consider the set of all rational numbers . Then it can be shown that for an arbitrary real number
*a*and desired accuracy , one can always find some rational number*q*such that . Hence the set of rational numbers are dense in the set of real numbers () - 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).