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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 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 of A in X is the union of A and the set of all limits of sequences of elements in A (its limit points),
Then A is dense in X if
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 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.
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.
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.
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.
- Nicolas Bourbaki (1989) . General Topology, Chapters 1–4. Elements of Mathematics. Springer-Verlag. ISBN 3-540-64241-2.
- Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) , Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446