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 $i\le j$ 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 $i\le j\in D$ 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 $A^{\mathrm{\complement }}$, 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, $x_k\in A^{\mathrm{\complement }}$ 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 $0,2,0,4,0,6,0,8,\mathrm{\dots },0,2n,0,\mathrm{\dots }$ has $0$ as a cluster point. But clearly $0$ is not a limit point, as the sequence diverges in $ℝ$.