frequently in

Recall that a net is a function $x$ from a directed set $D$ to a set $X$ . The value of $x$ at $i\in D$ is usually denoted by $x_{i}$ . Let $A$ be a subset of $X$ . We say that a net $x$ is frequently in $A$ if for every $i\in D$ , there is a $j\in D$ such that $i\leq j$ and $x_{j}\in A$ .

Suppose a net $x$ is frequently in $A\subseteq X$ . Let $E:=\{j\in D\mid x_{j}\in A\}$ . Then $E$ is a cofinal subset of $D$ , for if $i\in D$ , then by definition of $A$ , there is $i\leq j\in D$ such that $x_{j}\in A$ , and therefore $j\in E$ .

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 $A\subseteq X$ iff it is not frequently in $A^{\complement}$ , its complement. Suppose $x$ is eventually in $A$ . There is $j\in D$ such that $x_{k}\in A$ for all $k\geq j$ , or equivalently, $x_{k}\in A^{\complement}$ for no $k\geq j$ . The converse is can be argued by tracing the previous statements backwards.

In a topological space $X$ , a point $a\in X$ is said to be a cluster point of a net $x$ (or, occasionally, $x$ clusters at $a$ ) if $x$ is frequently in every neighborhood 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 $0,2,0,4,0,6,0,8,\ldots,0,2n,0,\ldots$ has $0$ as a cluster point. But clearly $0$ is not a limit point, as the sequence diverges in $\mathbb{R}$ .

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