Definition:Net (Preordered Set)

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Let $X$ be a nonempty set.

Let $\struct {\Lambda, \precsim}$ 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 $\struct {D, \le}$ 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 $\map \phi d = x_d$, and subsequently denote the net $\phi$ by $\family {x_d}_{d \mathop \in D}$, 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, $\struct {\set {a, b}, \le}$, in which the relation is $a \le a$ and $b \le b$ is a preorder, but not a directed set.

Also see

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.