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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".

[edit] 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.

[edit] Examples

  1. Consider the set of all rational numbers \scriptstyle \mathbb{Q}. 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 is 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).
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