Dense set

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.

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 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 {U_n} is a sequence of dense open sets in a complete metric space, X, then is also dense in X. This fact is one of the equivalent forms of 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 γM.

See also

• Isolated point
• Dense order
• Dense-in-itself
• Separable space, a space with a countable dense subset
• Nowhere dense set, the opposite notion