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Recently, I've been studying nets using Professor Munkres' Topology book. It defines a subnet as follows: Let $f\colon J \to$ X be a net in $X$; let $f(\alpha) = x_\alpha$. If $K$ is a directed set and $g \colon K \to J$ is a function such that:

  1. $i \le j \Rightarrow g(i) \le g(j)$.
  2. $g(K)$ is cofinal in $J$.

then the composite function $f \circ g \colon K \to$ X is called subnet.

My question is: Wouldn't it be enough to define a subnet as a restriction of $f$ to a cofinal subset of $J$?

Sebastiano
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Gleberson Antunes
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    No: for instance, you want to have that in a compact space every net has a convergent subnet. However, in the compact space $\beta \mathbb{N}$ there exist no non-trivial convergent sequences. Hence for a sequence $(x_n){n \in \mathbb{N}}$ of pairwise distinct elements of $\beta \mathbb{N}$ there does not exist a cofinal subset $N$ of the index set $\mathbb{N}$ such that $(x_n){n \in N}$ converges. – Ulli Nov 26 '22 at 19:50
  • Thanks for the answer @Ulli. I had thought that it would be enough to consider a subset K cofinal of J because, being K cofinal it is automatically directed, so f|K would be a net ...

    Anyway, I'll think about it more.

     – Gleberson Antunes Nov 26 '22 at 21:15

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