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, 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 a 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.
An alternative definition in case of the metric spaces is the following: A set A in a metric space X is dense if every [x] in [X] is a limit of a sequence of elements in A.
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 [\gamma M]
See also
- density (disambiguation)
- separable space, a space with a countable dense subset
- nowhere dense set, the opposite notion
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