Dense set

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 Failed to parse (Missing texvc executable; please see math/README to configure.): x

in X is a limit of a sequence of elements in A. That is, A is dense when 

Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{A} = X,

where Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{A}

denotes the  closure of A.

If Failed to parse (Missing texvc executable; please see math/README to configure.): \{U_n\}

is a sequence of dense open sets in a complete metric space, X, then Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): M

is dense in its completion Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma M

.

See also

• Dense order
• Dense-in-itself
• Separable space, a space with a countable dense subset
• Nowhere dense set, the opposite notioncs:Hustá množina

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