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; that is, the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or 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). 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; that is, the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or 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, 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 non-empty intersection with every non-empty 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. 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 A ¯ {displaystyle displaystyle {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), Then A is dense in X if Note that A ⊆ { lim n a n : ∀ n ≥ 0 ,   a n ∈ A } {displaystyle displaystyle A;subseteq ;{lim _{n}a_{n}:,forall n;geq ;0,, a_{n}in A}} . If { U n } {displaystyle displaystyle {U_{n}}} is a sequence of dense open sets in a complete metric space, X, then ⋂ n = 1 ∞ U n {displaystyle displaystyle igcap _{n=1}^{infty }U_{n}} is also dense in X. This fact is one of the equivalent forms of the Baire category theorem. 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 complex-valued continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. In other words, the polynomial functions are dense in the space C of continuous complex-valued functions on the interval , equipped with the supremum norm. Every metric space is dense in its completion.