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 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 <math>M<math> is dense in its completion <math>\gamma M<math>
See also
- separable space, a space with a countable dense subset
- nowhere dense set, the opposite notionja:稠密