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 one 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.

Density in metric spaces

An alternative definition of dense set in the case of metric spaces is the following: The set A in a metric space X is dense if every $x$ in X is a limit of a sequence of elements in A. That is, A is dense when

bar{A} = X,

where $bar{A}$ denotes the closure of A. If ${U_n}$ 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 allows one to easily prove the Baire category theorem.

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 $M$ is dense in its completion $gamma M$ .

Dense set

Density in metric spaces

Examples

See also