There are loads of questions about best references for topology. However, I'm posting this one because I'm sure it is quite different.
I have started reading Bourbaki General Topology I and I feel comfortable with the first 5 sections. Maybe Bourbaki's Theory of Sets has too much weight sometimes (for example he focuses on canonical decompositions too much in my opinion). But I think the book introduces the topological concepts in a very simple way, saying a lot with few words. However, sections 6 starts with filters. I could skip this section but some other definitions use filters so…
Hence, I decided to look at othere references. In particular Kuratowski's books. But the treatment is horrible in my opinion. I also found Degundji and Willard. They seems good. Moreover, Degundji has an approach near differential and algebraic topology, which is the topology I'm interested in.
So my question(s) is(are): Are filters avoidable in topology or we need it to define some concepts? If they are necessary, should I continue with Bourbaki's book or move to one of the others?
Thanks
Best Answer
My advice: learn both filters and nets, for example from Engelking's "General Topology": Section 1.6 and Problems 1.7.18--21. This way you will know both approaches and you will be able to choose whichever is better suited to the problem at hand. As mentioned in the comments filters have an easy counterpart for `subsequence'; on the other hand iterated limits are easier to grasp using nets.