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 members 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 if the smallest closed subset itself.

The is the least cardinality

Definition

of a topological space is said to be a if any of the following equivalent conditions are satisfied: The smallest closed subset itself. The closure The interior of the complement or is a limit point every neighborhood intersects

is a basis then this list can be extended to include: every neighborhood intersects

Density in metric spaces

An alternative definition of dense set in the case of metric spaces is the following. When the topology is given by a metric, the closure is the union and the set of all limits of sequences (its limit points

is a sequence of dense open This fact is one of the equivalent forms of the Baire category theorem.

Examples

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. The irrational numbers are another dense subset which shows that a topological space may have several disjoint dense subsets (in particular, two dense subsets may be each other's complements), and they need not even be of the same cardinality. 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. The intersection of two dense open subsets of a topological space is again dense and open. The empty set is a dense subset of itself. But every dense subset of a non-empty space must also be non-empty.

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 equipped with the supremum norm.

Every metric space is dense in its completion.

Properties

Every topological space equipped with the discrete topology equipped with the trivial topology is dense, and every topology for which every non-empty subset is dense must be trivial.

Denseness is transitive (in the respective subspace topology

The image of a dense subset under a surjective continuous function is again dense. The density of a topological space (the least of the cardinalities of its dense subsets) is a topological invariant.

A topological space with a connected dense subset is necessarily connected itself.

Continuous functions into Hausdorff spaces into a Hausdorff space

For metric spaces there are universal spaces, into which all spaces of given density can be embedded the space of real continuous functions on the product copies of the unit interval.

Related notions

is called a limit point itself, and an isolated point otherwise. A subset without isolated points is said to be dense-in-itself.

is called nowhere dense is dense. Equivalently, a subset of a topological space is nowhere dense if and only if is called meagre. The rational numbers, while dense in the real numbers, are meagre as a subset of the reals.

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.

An embedding as a dense subset of a compact space is called a compactification

A linear operator between topological vector spaces is said to be densely defined if its domain and if its range See also Continuous linear extension.

is hyperconnected A topological space is submaximal if and only if every dense subset is open.

See also

References

proofs

General references