I was wondering how popular is the use of filters at a master level?
If you were a student with no acknowledge of the concept, would you start the reading of an interesting book using this concept (provided there is an appendix explaining it)?
Thank you for sharing your points of view.
In my experience most topological spaces that appear in a module whose focus is not purely topological are metric or otherwise simple enough that using (ultra)filters to describe their properties is overkill. Usually sequences is enough.
Thus you want your topology modules to build up some familiarity with sequences to make them applicable to the non-topology modules. And then it seems like a waste of time to reformulate things as filters and reteach the same material.
While ultrafilters might make a proof of Tychonoff's theorem straightforward, you don't necessarily save time if you have no other reason to cover them.
The main reason I see to use (ultra)filters is they come up somewhere else in the subject matter. For example Boolean algebras, Models, or Stone-Cech compactifications. To understand these things you need to be familiar with filters. And if that understanding is already forthcoming there are no disadvantages to using them to describe the topology as well.