Definition:Net (Preordered Set)

Definition
Let $X$ be a nonempty set.

Let $\left({\Lambda, \precsim}\right)$ be a preordered set.

Let $F: \Lambda \to X$ be a mapping.

Then $f$ is referred to as a net.

Other Definition
Let $X$ be a set.

Let $\left({D, \le}\right)$ be a directed set.

A mapping $\phi: D \to X$ from $D$ to $X$ is called a net in $X$.

It is common to write $\phi \left({d}\right) = x_d$, and subsequently denote the net $\phi$ by $\left({x_d}\right)_{d \in D}$, mimicking the notation for indexed sets and sequences.

The first definition is not equivalent to this one because a directed set is more than a preordered set.

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

Also see

 * Definition:Convergent Net


 * Definition:Moore-Smith Sequence

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