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 every point x in X either belongs to A or is a limit point of A.^{[1]} 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 (i.e., A has nonempty intersection with every nonempty open subset of X). 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.
Contents
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
Note that . If is a sequence of dense open sets in a complete metric space, X, then 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 (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 nonempty open set.
By the Weierstrass approximation theorem, any given complexvalued 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 complexvalued functions on the interval [a,b], equipped with the supremum norm.
Every metric space is dense in its completion.
Properties
Every topological space is a dense subset of itself. For a set X equipped with the discrete topology the whole space is the only dense set. Every nonempty subset of a set X equipped with the trivial topology is dense, and every topology for which every nonempty 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 (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 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.
For metric spaces there are universal spaces, into which all spaces of given density can be embedded: a metric space of density α is isometric to a subspace of C([0,1]^{α}, R), the space of real continuous functions on the product of α copies of the unit interval. ^{[2]}
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 denseinitself.
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 κresolvable 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
References
Notes
 ^ Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 048668735X
 ^ Kleiber, Martin; Pervin, William J. (1969). "A generalized BanachMazur theorem". Bull. Austral. Math. Soc. 1: 169–173. doi:10.1017/S0004972700041411.
General references
 Nicolas Bourbaki (1989) [1971]. General Topology, Chapters 1–4. Elements of Mathematics. SpringerVerlag. ISBN 3540642412.
 Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: SpringerVerlag, ISBN 9780486687353, MR 507446