Dense set
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). Formally, formula_1 is dense in formula_2 if the smallest closed subset of formula_2 containing formula_1 is formula_2 itself. The of a topological space  formula_2 is the least cardinality of a dense subset of formula_7 Definition. A subset formula_1 of a topological space formula_2 is said to be a of formula_2 if any of the following equivalent conditions are satisfied: and if formula_11 is a basis of open sets for the topology on formula_2 then this list can be extended to include: Density in metric spaces. An alternative definition of dense set in the case of metric spaces is the following. When the topology of formula_2 is given by a metric, the closure formula_14 of formula_1 in formula_2 is the union of formula_1 and  the set of all limits of sequences of elements in formula_1 (its "limit points"), formula_19 Then formula_1 is dense in formula_2 if formula_22 If formula_23 is a sequence of dense open sets in a complete metric space, formula_24 then formula_25 is also dense in formula_7 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 formula_27 can be uniformly approximated as closely as desired by a polynomial function. In other words, the polynomial functions are dense in the space formula_28 of continuous complex-valued functions on the interval formula_29 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 formula_2 equipped with the discrete topology, the whole space is the only dense subset. Every non-empty subset of a set  formula_2 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 formula_32 and formula_33 of a topological space formula_2 with formula_35 such that formula_1 is dense in formula_37 and formula_37 is dense in formula_33 (in the respective subspace topology) then formula_1 is also dense in formula_41 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 formula_42 into a Hausdorff space formula_43 agree on a dense subset of formula_2 then they agree on all of formula_7 For metric spaces there are universal spaces, into which all spaces of given density can be embedded: a metric space of density formula_46 is isometric to a subspace of formula_47 the space of real continuous functions on the product of formula_46 copies of the unit interval. Related notions. A point formula_49 of a subset formula_1  of a topological space formula_2 is called a limit point of formula_1 (in formula_2) if every neighbourhood of formula_49 also contains a point of formula_1 other than formula_49 itself, and an isolated point of formula_1 otherwise. A subset without isolated points is said to be dense-in-itself. A subset formula_1 of a topological space formula_2 is called nowhere dense (in formula_2) if there is no neighborhood in formula_2 on which formula_1 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 formula_24 a subset formula_1 of formula_2 that can be expressed as the union of countably many nowhere dense subsets of formula_2 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 of a topological space formula_2 as a dense subset of a compact space is called a compactification of formula_7 A linear operator between topological vector spaces formula_2 and formula_43 is said to be densely defined if its domain is a dense subset of formula_2 and if its range is contained within formula_72 See also Continuous linear extension. A topological space formula_2 is  hyperconnected if and only if every nonempty open set is dense in formula_7 A topological space is submaximal if and only if every dense subset is open. If formula_75 is a metric space, then a non-empty subset formula_43 is said to be formula_77-dense if formula_78 One can then show that formula_79 is dense in formula_75 if and only if it is ε-dense for every formula_81 References. proofs