I recently had a look into general topology and now I am trying to wrap my head around the notion of a net. I understand its definition as a map from an upward directed set into our topological space, but I do not get, why it has to be this general. As I see it, since we want to characterize our topology by the convergence of its nets, it would be preferable to choose assumptions on our index set as strong as possible in order to have fewer nets we need to work with, while still having as much as needed to be able to transfer the theorems for sequences in metric spaces to the more general setting of a topological space. For example now, after reading the proofs of some of those theorems, to me lattice-indexed nets would appear as a more natural choice of definition, since most of the nets I encountered used families of neighbourhoods as index sets and so, if I am not mistaken, all the proofs would still work. So my questions now are: Is there a theorem that actually needs nets in there most general definition? Or maybe is there just no practical difference between the two? Or is there something completely different I missed?
(1.) There wouldn't be much practical difference, and (2.) If in a certain proof, you would want to construct a net and to VERIFY that it is indeed a net, then you would need to do much less verification work if your definition is the most general possible, yet that captures the essence of convergence and applies to all topological spaces. So, from this (amount of verification work) point of view, it does not seem justified to choose that we do want the strongest assumptions on our index set.