Introduction to Filters in Topology
Filters and ultrafilters play a basic role in topology: they allow for a general theory of convergence in topological spaces when sequences do not suffice. Moreover, they do so in an arguably somewhat more graceful and canonical way than the rival theory of nets: see my notes on convergence where I develop nets, then filters, then compare the two. But either way the theory of convergence in topological spaces is a sort of "finished product" -- it is there for you to use, and it is certainly useful, but the theory itself and how to apply it seem to be by now rather thoroughly understood: for instance, the theory of convergence stands at almost exactly the same point today as it did after the publication of Bourbaki's Topologie Générale by 1950. (Or so is my impression. Please let me know if I am simply speaking from ignorance.)
On the other hand, the use of (especially ultra-)filters in mathematics generally is definitely on the rise, especially in recent years. The fundamental construction here is the ultraproduct of structures. Its importance in model theory is guaranteed by the theorem of Jerzy Łoś. Very roughly, the ultraproduct of a family of structures inherits all the elementary properties shared by "most" of the structures in the family while at the same time being very conveniently "large" (in more model-theoretic terminology, ultraproducts enjoy good saturation properties). This leads swiftly to the idea of nonstandard models, which were initially most popular in analytic contexts. But nonstandard methods seem not to have really caught on with the majority of the analysis community. It is not so in algebra, algebraic geometry and number theory, where "nonstandard methods" are becoming increasingly, um, standard. My guess is that 10-20 years from now, ultrafilters will be a very familiar topic to graduate students, because of their model-theoretic applications.
To get just a little taste of the use of ultrafilters in model theory, you could read the final chapter of the lecture notes for the short summer course on model theory I taught in 2010. (If you don't know any model theory already, you'll need to look at the earlier chapters as well to really understand it, but perhaps you can gain an impression of what's going on just by skimming this chapter.) But that's barely the tip of the iceberg: there are whole books written on this subject, e.g. by (Bell and Slomson) and (Chang and Keisler).
Finally, the study of saturation properties of ultraproducts leads quickly to some nontrivial set-theoretic considerations. The fact that model theorists know and care much more about set theory than many other working mathematicians can probably be traced to this, at least in part.
Bourbaki's General Topology, Volume 1 devotes a chapter to filters. The notion of net and equivalence between filters and nets is developed (partly in a set of guided exercises) in Kelley's General Topology. The use of ultraproducts in Commutative algebra by Schoutens might interest you. The theory of ultrafilters by Comfort and Negrepontis is encyclopedic. Jech's Set theory has a chapter on ultrafilters. Sections 4 and 6 of Chang and Keisler's Model Theory is also worth a look. These are all available on the website library.nu.
As far as online resources:
- Notes on convergence by Pete Clark,
- Blog post by Todd Trimble,
- Blog posts by Terence Tao (see also the posts tagged ultralimit-analysis and nonstandard-analysis)
I am also writing a series of blog posts (unfinished) on the subject.