Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A - for instance, every real number is either a rational number or has one arbitrarily close to it (see Diophantine approximation).

Formally, a subset A of a topological space X is dense in X if for any point x in X, any neighborhood of x contains at least one point from A. Equivalently, A is dense in X if and only 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.

The density of a topological space X is the least cardinality of a dense subset of X.

Density in metric spaces

An alternative definition of dense set in the case of metric spaces is the following. When the topology of X is given by a metric, the closure $\overline{A}$ of A in X is the union of A and the set of all limits of sequences of elements in A (its limit points),

$\overline{A} = A \cup \{ \lim_n a_n : \forall n \ge 0, \ a_n \in A \}.$

Then A is dense in X if

$\overline{A} = X.$

Note that $A \subseteq \{ \lim_n a_n : \forall n \ge 0, \ a_n \in A \}$. If $\{U_n\}$ is a sequence of dense open sets in a complete metric space, X, then $\cap^{\infty}_{n=1} U_n$ is also dense in X. 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.

By the Weierstrass approximation theorem, any given complex-valued continuous function defined on a closed interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. In other words, the polynomial functions are dense in the space C[a,b] of continuous complex-valued functions on the interval [a,b], equipped with the supremum norm.

Every metric space is dense in its completion.

Properties

Every topological space is dense in itself. For a set X equipped with the discrete topology the whole space is the only dense set. Every non-empty subset of a set X equipped with the trivial topology is dense, and every topology for which every non-empty subset is dense must be trivial.

Denseness is transitive: Given three subsets A, B and C of a topological space X with A ⊆ B ⊆ C such that A is dense in B and B is dense in C (in the respective subspace topology) then A is also dense in C.

The image of a dense subset under a surjective continuous function is again dense. The density of a topological space is a topological invariant.

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

Continuous functions into Hausdorff spaces are determined by their values on dense subsets: if two continuous functions f, g : X → Y into a Hausdorff space Y agree on a dense subset of X then they agree on all of X.

Related notions

A point x of a subset A of a topological space X is called a limit point of A (in X) if every neighbourhood of x also contains a point of A other than x itself, and an isolated point of A otherwise. A subset without isolated points is said to be dense-in-itself.

A subset A of a topological space X is called nowhere dense (in X) if there is no neighborhood in X on which A is dense. Equivalently, a subset of a topological space is nowhere dense if and only if the interior of its closure is empty. The interior of the complement of a nowhere dense set is always dense. The complement of a closed nowhere dense set is a dense open set. Given a topological space X, a subset A of X that can be expressed as the union of countably many nowhere dense subsets of X 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 if it contains κ pairwise disjoint dense sets.

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

A linear operator between topological vector spaces X and Y is said to be densely defined if its domain is a dense subset of X and if its range is contained within Y. See also continuous linear extension.

A topological space X is hyperconnected if and only if every nonempty open set is dense in X. A topological space is submaximal if and only if every dense subset is open.

See also

• Dense order

References

General references

• Nicolas Bourbaki (1989) [1971]. General Topology, Chapters 1–4. Elements of Mathematics. Springer-Verlag. ISBN 3-540-64241-2. 
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446