# Denseness

Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
Citable Version  [?]

This editable Main Article is under development and subject to a disclaimer.

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 ${\displaystyle \scriptstyle A\subset X}$ 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 ${\displaystyle \scriptstyle {\overline {A}}=X}$); equivalently, the only closed set in X containing A is X itself.

## Examples

1. Consider the set of all rational numbers ${\displaystyle \scriptstyle \mathbb {Q} }$. Then it can be shown that for an arbitrary real number a and desired accuracy ${\displaystyle \scriptstyle \epsilon >0}$, one can always find some rational number q such that ${\displaystyle \scriptstyle |q-a|<\epsilon }$. Hence the set of rational numbers are dense in the set of real numbers (${\displaystyle \scriptstyle {\overline {\mathbb {Q} }}=\mathbb {R} }$)
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).