I have learned point-set topology using filters. Now I do functional analysis where we are using nets to do topological stuff. Therefore I search an introductory text on nets that is suitable for this purpose, i.e. to lay the foundations for usage in FA and maybe that the text assumes some knowledge in point-set topology, I don't want to start from scratch.
One of the very best references that I’ve seen is a PDF, Translating Between Nets and Filters, by Saitulaa Naranong that can still be found here. (I’m aware of one typo: $\Phi$ and $\Psi$ have been interchanged in the displayed implication at the top of page $11$. The one-sentence paragraph two lines down (‘In other words ...’) is correct.)
- Kelley, General Topology, which popularized nets and the terminology, and
- Pete Clark, Convergence, a paper developing the theory of sequences, filters and nets, and proving implications and equivalences between them.
A caveat to my answer: I don't know functional analysis, so this is just what has helped me in understanding point set topology better. I find nets build intuition in point set topology.
In no particular order:
- Try the section in Engelking's General Topology starting on p. 49.
- Munkres's Topology has a little section on nets with some good exercises.
- Willard's General Topology chapter 4 is good.
I recommend learning the canonical way to translate nets into filters, and vice-versa. If you still want more practice, I recommend going through some of the proofs in point set topology (e.g. tube lemma) in the context of nets, and then the corresponding one in terms of filters. It's fun!
I don't know a reference that explains nets for functional analysis, but here is an introductory book on nets: Limits - A New Approach to Real Analysis by Alan F. Beardon.