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, intuitively, any point in X can be "well-approximated" by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of x contains at least a point from A.

Equivalently, A is dense in X 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.

An alternative definition in case of the metric spaces is the following: A set A in a metric space X is dense if every <math>x</math> in <math>X</math> is a limit of a sequence of elements in A.

Examples

• every topological space is dense in itself
• the real numbers with the usual topology have the rational numbers and the irrational numbers as dense subsets
• a metric space <math>M</math> is dense in its completion <math>\gamma M</math>

See also

• separable space, a space with a countable dense subset
• nowhere dense set, the opposite notioncs:Hustá množina

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