\begin{definition}[Definition:Convergent Net]
Let $\struct {X, \tau}$ be a topological space.
Let $\struct {I, \le}$ be a directed set.
Let $\family {x_i}_{i \mathop \in I}$ be a net.
$\family {x_i}$ is said to '''converge to''' $x \in X$, denoted $\ds x_i \to x$ 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 $\family {x_i}$'''.
A net $\family {x_i}_{i \mathop \in I}$ 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'''.
\end{definition}