Dense set
From Wikipedia, the free encyclopedia
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.
[edit] 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
where 
denotes the closure of A. If {Un} is a sequence of dense open sets in a complete metric space, X, then 
is also dense in X. This fact allows one to easily prove the Baire category theorem.
[edit] 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.
[edit] See also
- Dense order
- Dense-in-itself
- Separable space, a space with a countable dense subset
- Nowhere dense set, the opposite notion
