Definition:Subnet

Definition
Let $\left({E, \le}\right)$ and $\left({D,\preceq}\right)$ be directed sets.

Let $A$ be a non-empty set.

Let $T: E \to A$ and $S: D \to A$ be nets in $A$.

Then $T$ is a subnet of $S$ iff there exists a cofinal mapping $N: E \to D$ such that $T = S \circ N$.

That is, for each $m$ in $D$, there is an $n$ in $E$ such that for each $p \in E$, $p \ge n \implies N \left({p}\right) \succeq m$.

Also defined as
Some authors require the mapping $N$ to be monotone, while others do not.