Definition:Net (Preordered Set)
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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
Let $X$, be a set, and let $\left({D, \leq}\right)$ 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, $\left({ \left\{{a, b}\right\}, \leq }\right)$, in which the relation is $a\leq a$ and $b\leq b$ is a preorder, but not a directed set.
