Filters vs nets in topology
what are (dis)advantages of the net vs filter languages.
nets:
- Some statements are easier with nets. e.g.
If $X$ is a topological space and $A\subset X$ then $a\in \overline A$ iff some net on $A$ converges to $a$.
with filters one has to define what convergence of a filter on $A$ in $X$ means.
- Because nets are more intuitive and we can think of a net as a collection of points (somehow ordered), some natural questions may be inspired; for example:
Suppose $X$ is a topological space and every net on $A\subseteq X$ has a convergent subnet. Is $\overline A$ compact?
(the answer is yes in Tychonoff spaces)
Almost all statements about sequences in analysis, can be translated to nets on topological or uniform spaces. e.g. this or this or even this with nets.
Net theorems will stick in mind, especially if you have studied analysis, because they can be imagined. But filters are more abstract. This is why I prefer nets.
filters:
With filters some proofs about compactness are easier. Even Tychonoff Theorem can be proved with filters.
Any (diagonal) uniformity is a filter. Before studying uniform spaces one should study filters.
Filter has something to do with Bornology.
Convergence of a filter controls the convergence of all nets which correspond to that filter. This says filters only have the necessary features for convergence while nets have features that are hardly pertinent to convergence.
Unlike superfilters, there are several definitions for subnets. So before using the word subnet you should clarify what you mean by that.
More about the so-called equivalence of filters and nets can be found in last pages of this pdf.
Nets involve a partial order relation on the indexing set, and only a part of the information contained in that relation is relevant for topological purposes. The relevant part is just what is retained when one passes from the net to the associated filter. So, in a sense, the use of filters discards irrelevant information that is present in nets.
I believe I learned about nets before filters, so my preference for filters is probably not based on timing. It's more likely to have resulted from a (congenital?) preference for simplicity and for discarding or at least ignoring irrelevant information. I agree, though, that after one learns basic notions in the context of sequences, nets, being rather similar to sequences, will be more intuitive, until one encounters subnets.
Filters are very natural. Filters tell you when something happens "almost everywhere", that is on a "big" set.
Convergence is something that needs to happen "almost everywhere", that is, $x_i\to x$ (where $x_i$ is a net) if every open set contains "almost all" the $x_i$'s. That's a very obvious use of "almost everywhere". In some sense, almost all the net is almost everywhere around $x$.
This is why filters are great for convergence.