Introduction to Filters in Topology Question: What are some good resources for a student who has taken algebraic and point-set topology and who wishes to learn how filters and ultrafilters are applied in topology?

Motivation: I've seen in a number of papers where ultrafilters are briefly mentioned as tools which can be used to generalize an argument or something, but the few papers I've found which introduce filters mainly use them as as tools in Logic and mention topology only in passing. The Wikipedia page gives some good starting points, but I felt that it was more of a reference list than an introduction to the topic.
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.
Also consider my book. It is not a standard course on general topology, but rather a generalization of general topology.

In my theory all general topology is based on filters (and filters are studied in my book in details, making it definitely the world best reference on the topic of filters).

In fact, it is not general topology, but its generalization. However, limit, connectedness, continuity, limit (even generalized limit of discontinuous function), etc. appear in my theory just like as they appear in standard general topology. And these concepts are expressed by nice algebraic formulas, not as a mess of quantifiers as in any other topology book.

Curious fact: Continuity generalizes for multivalued functions in three different ways.

My theory applies both to calculus and to discrete mathematics.

You can study my theory after or before legacy general topology. The book even contains an introduction for beginning students who know only basic calculus and basic set theory.