Definition:Subnet

Definition

Let $\struct {E, \le}$ and $\struct {D, \preceq}$ 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 $S$ if and only if 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 $p \in E$, $p \ge n \implies \map N p \succeq m$.

Also defined as

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

Sources

• 1955: John L. Kelley: General Topology: Chapter $2$: Subnets and Cluster Points