Subnet

Last updated Nov 6, 2022

Let $X$ be a Set and let ${x_{\alpha}}{\alpha \in A} \subset X$ be a Net. We say a Net ${y{\beta}}{\beta \in B} \subset X$ is a Subnet of ${x{\alpha}}_{\alpha \in A}$ if $\exists n: B \to A$ s.t.

1. $n$ is an Order-Preserving Function.
2. $n$ is a Final Function.
3. $y_{\bullet} = x_{\bullet} \circ n$. That is $y_{\beta}= x_{n(\beta)}$ for all $\beta \in B$.

1. This definition allows us to repeat elements. $n$ need not be injective.
2. This is the definition put forward by Willard in 1970.

• Index Set