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

Examples

1. Consider the set of all rational numbers $\scriptstyle \mathbb {Q}$. Then it can be shown that for an arbitrary real number a and 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] (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).