Definition:Net (Preordered Set)

Definition
Let $\mathcal{X}$ be a nonempty set and $\Lambda$ a preordered set (endowed with a preorder $\leq$) Any mapping from $\Lambda$ to $\mathcal{X}$ is called a net.

Note : Nets are extensions of sequences. In fact, a sequence over a set $\mathcal{X}$ is a mapping from $\N$ to $\mathcal{X}$ and $\N$ - endowed with the standard comparison relation $\leq$ (which is a partial order and a fortiori a preorder) - is a preordered set. Hence a sequence is a special case of a net.

Other Definition
A net in a set $X$ is a mapping from any directed set $D$ to $X$. The first definition is not equivalent to this one because a directed set is more than a preorder.

For example $(\{a,b\},\leq)$ in which the relation is $a\leq a$ and $b\leq b$ is a preorder, but not a directed set.

If $(X,\tau)$ is a topological space, we denote the net $\phi:(D,\leq)\to X$ as a "sequence" $\left\langle{\phi(d)}\right\rangle_{d\in D}=\left\langle{x_d}\right\rangle_{d\in D}$. We say that the net converges and denote it as $x_d\to x$ if for any open neighborhood $N$ of $x$ there exists $d_0\in D$ such that $\forall d\geq d_0;$ $x_d\in N$.