Dense set
In topology and related areas of mathematics a subset A of a topological space X is called 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. Equivalently, every nonempty open subset of X intersects A, or in other words: the interior of the complement of A is empty.
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
- separable space, a space with a countable dense subset
- nowhere dense set, the opposite notion