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. 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. 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.
Read more about Dense Set: Density in Metric Spaces, Examples, Properties, Related Notions
Famous quotes containing the words dense and/or set:
“In the dense light of wakened flesh
animal man is a prince. As from alabaster
a lucency animates him from heel to forehead.
Then his shadows are deep and not gray.”
—Denise Levertov (b. 1923)
“Nothing in medieval dress distinguished the child from the adult. In the seventeenth century, however, the child, or at least the child of quality, whether noble or middle-class, ceased to be dressed like the grown-up. This is the essential point: henceforth he had an outfit reserved for his age group, which set him apart from the adults. These can be seen from the first glance at any of the numerous child portraits painted at the beginning of the seventeenth century.”
—Philippe Ariés (20th century)