Dense set

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In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if any point x in X belongs to A or is a limit point of A.[1]

Formally, a subset A of a topological space X is dense in X if for any point x in X, any neighborhood of x contains at least one point from A. Equivalently, A is dense in X if and only if the only closed subset of X containing A is X itself. This can also be expressed by saying that the closure of A is X, or that the interior of the complement of A is empty.

The density of a topological space X is the least cardinality of a dense subset of X.

Contents

Density in metric spaces

An alternative definition of dense set in the case of metric spaces is the following. When the topology of X is given by a metric, the closure \overline{A} of A in X is the union of A and the set of all limits of sequences of elements in A (its limit points),

Then A is dense in X if

If {Un} is a sequence of dense open sets in a complete metric space, X, then \cap^{\infty}_{n=1} U_n is also dense in X. This fact is one of the equivalent forms of the Baire category theorem.

Examples

The real numbers with the usual topology have the rational numbers as a countable dense subset which shows that the cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The irrational numbers are another dense subset which shows that a topological space may have several disjoint dense subsets.

By the Weierstrass approximation theorem, any given complex-valued continuous function defined on a closed interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. In other words, the polynomial functions are dense in the space C[a,b] of continuous complex-valued functions on the interval [a,b], equipped with the supremum norm.

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