frequently in

Recall that a net is a function x from a directed setMathworldPlanetmath D to a set X. The value of x at iD is usually denoted by xi. Let A be a subset of X. We say that a net x is frequently in A if for every iD, there is a jD such that ij and xjA.

Suppose a net x is frequently in AX. Let E:={jDxjA}. Then E is a cofinal subset of D, for if iD, then by definition of A, there is ijD such that xjA, and therefore jE.

The notion of “frequently in” is related to the notion of “eventually in” in the following sense: a net x is eventually in a set AX iff it is not frequently in A, its complementPlanetmathPlanetmath. Suppose x is eventually in A. There is jD such that xkA for all kj, or equivalently, xkA for no kj. The converseMathworldPlanetmath is can be argued by tracing the previous statements backwards.

In a topological spaceMathworldPlanetmath X, a point aX is said to be a cluster pointPlanetmathPlanetmath of a net x (or, occasionally, x clusters at a) if x is frequently in every neighborhoodMathworldPlanetmathPlanetmath of a. In this general definition, a limit point is always a cluster point. But a cluster point need not be a limit point. As an example, take the sequence 0,2,0,4,0,6,0,8,,0,2n,0, has 0 as a cluster point. But clearly 0 is not a limit point, as the sequence diverges in .

Title frequently in
Canonical name FrequentlyIn
Date of creation 2013-03-22 17:14:23
Last modified on 2013-03-22 17:14:23
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 03E04
Synonym clusters at
Defines cluster point of a net