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 and $\scriptstyle A\subset X$, then A is said to be dense in X, or is a dense set in X, if the closure of A coincides with X. That is, $\scriptstyle {\overline {A}}=X$.

Examples

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

2. The set of algebraic polynomials can uniformly approximate any continuous function on a fixed interval [a,b] (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 set of continuous functions on the interval [a,b] (with respect to the uniform topology).