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 smallest 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.

Technology Types

general topology

Synonyms

DenseDense subsetDense subspaceSequentially dense

Translations

Conjunt dens (ca)Conjunto denso (es)Conjunto denso (pt)Densa aro (eo)Dichte Teilmenge (de)Hustá množina (cs)Insieme denso (it)Partie dense (fr)Tät mängd (sv)Zbiór gęsty (pl)Плотное множество (ru)Щільна множина (uk)مجموعة كثيفة (ar)조밀 집합 (ko)稠密集 (zh)稠密集合 (ja)