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 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.

[edit] 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 $\bar{A}$ of A in X is the union of A and the set of all limits of sequences of elements in A (its limit points),

$\bar{A} = A \bigcup \{ \lim_n a_n : \forall n \ge 0, \ a_n \in A \}.$

Then A is dense in X if

$\bar{A} = X.$

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 is one of the equivalent forms of the Baire category theorem.