Dense set

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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, ${\displaystyle A}$ is dense in ${\displaystyle X}$ if the smallest closed subset of ${\displaystyle X}$ containing ${\displaystyle A}$ is ${\displaystyle X}$ itself.^[1]

The density of a topological space ${\displaystyle X}$ is the least cardinality of a dense subset of ${\displaystyle X.}$