Dense and Nondense Sets

concepts in set theory. A set E is said to be dense in M if every point of the set M is a limit point of E—that is, if every neighborhood in M contains a point of E. Sets dense relative to the entire line are said to be everywhere dense. A set is nondense, or nowhere dense, relative to the line if it is dense in no interval—in other words, if every interval of the line contains a subinterval that is entirely free of points of the given set. Sets that are nondense in the plane or, in general, in an arbitrary topological space are defined in a similar manner.

A closed set is nondense if, and only if, its complement is everywhere dense. An example of a closed nondense set is the Cantor ternary set, which is a perfect set. A set that is a countable union of nowhere dense sets is called a set of first category, and the complement of a set of first category is a set of second category. These concepts play an important role in the theory of normed linear spaces. Different categories of sets are also of importance in the theory of the uniqueness of trigonometric series.