Definition:Convergent Net

Definition
Let $\left({X, \tau}\right)$ be a topological space, and let $\left({I, \leq}\right)$ be a directed set.

Let $\left({x_i}\right)_{i \in I}$ be a net.

The net $\left({x_i}\right)$ is said to converge to $x \in X$, denoted $\displaystyle x_i \to x$ or $\lim x_i = x$, iff:


 * $\forall U \in \tau: x \in U \implies \exists i_0 \in I: \forall i \ge i_0: x_i \in U$

That is, for every open $U$ with $x \in U$, there exists an $i_0 \in I$ such that forall $i \ge i_0$, $x_i \in U$.

If $x_i \to x$, then $x$ is called a limit (point) of $\left({x_i}\right)$.

A net $\left({x_i}\right)_{i \in I}$ is called convergent if there is an $x \in X$ such that $x_i \to x$.

If such an $x$ does not exist, the net is said to be divergent.

Cluster Point
The net $\left({x_i}\right)$ is said to cluster at $x \in X$, denoted $\displaystyle x_i \mathop{\longrightarrow}_{\text{cl}} x$, iff:


 * $\forall U \in \tau, i_0 \in I: x \in U \implies \exists i \ge i_0: x_i \in U$

That is, for every open $U$ with $x \in U$, and for every $i_0 \in I$, there is an $i \ge i_0$ such that $x_i \in U$.

If $\displaystyle x_i \mathop{\longrightarrow}_{\text{cl}} x$, then $x$ is called a cluster point of $\left({x_i}\right)$.