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 $i\le j$ and $x_j\in A$

Suppose a net $x$ is frequently in $A\subseteq X$ Let $E:=\lbrace j\in D\mid x_j\in A\rbrace$ Then $E$ is a cofinal subset of $D$ for if $i\in D$ then by definition of $A$ there is $i\le j\in D$ 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 $k\ge j$ or equivalently, $x_k\in A^{\complement}$ for no $k\ge j$ 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 $\mathbb{R}$