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Denseness

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This is a draft article, under development. These unapproved articles are 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 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).

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