In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
is a basis of open sets for the topology on
then this list can be extended to include: An alternative definition of dense set in the case of metric spaces is the following.
and the set of all limits of sequences of elements in
is a sequence of dense open sets in a complete metric space,
This fact is one of the equivalent forms of the Baire category theorem.
The real numbers with the usual topology have the rational numbers as a countable dense subset which shows that the cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself.
Perhaps even more surprisingly, both the rationals and the irrationals have empty interiors, showing that dense sets need not contain any non-empty open set.
[proof 1] The empty set is a dense subset of itself.
By the Weierstrass approximation theorem, any given complex-valued continuous function defined on a closed interval
can be uniformly approximated as closely as desired by a polynomial function.
In other words, the polynomial functions are dense in the space
of continuous complex-valued functions on the interval
Every metric space is dense in its completion.
equipped with the discrete topology, the whole space is the only dense subset.
equipped with the trivial topology is dense, and every topology for which every non-empty subset is dense must be trivial.
Continuous functions into Hausdorff spaces are determined by their values on dense subsets: if two continuous functions
the space of real continuous functions on the product of
A subset without isolated points is said to be dense-in-itself.
Equivalently, a subset of a topological space is nowhere dense if and only if the interior of its closure is empty.
that can be expressed as the union of countably many nowhere dense subsets of
A topological space with a countable dense subset is called separable.
A topological space is a Baire space if and only if the intersection of countably many dense open sets is always dense.
A topological space is called resolvable if it is the union of two disjoint dense subsets.
More generally, a topological space is called κ-resolvable for a cardinal κ if it contains κ pairwise disjoint dense sets.
as a dense subset of a compact space is called a compactification of
A linear operator between topological vector spaces
is hyperconnected if and only if every nonempty open set is dense in
A topological space is submaximal if and only if every dense subset is open.
is a metric space, then a non-empty subset