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?


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    $\begingroup$ Filters. Are. Awesome. $\endgroup$ – Asaf Karagila Sep 21 '18 at 8:37
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    $\begingroup$ You can avoid filters by using the equivalent concept of nets. Filters are very powerful though. Tychonoff's theorem almost comes for free by developping filter theory... $\endgroup$ – user370967 Sep 21 '18 at 10:03
  • $\begingroup$ @AsafKaragila. I don't think the same. At least after reading yesterday some pages about filters in Bourbaki's book. $\endgroup$ – Dog_69 Sep 21 '18 at 16:07
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    $\begingroup$ Sure, but the first time I tasted whisky, I also didn't think it was that great. Then again, it was probably something like Johnny Walker Red Label, or some other awful thing. You need to develop both the taste, and see elegance in the usage. I don't know if Bourbaki will deliver on either, I haven't read it. $\endgroup$ – Asaf Karagila Sep 21 '18 at 16:09
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    $\begingroup$ filters vs nets are easily seen to be equivalent approaches. Filters are often more convenient (as it's easy to talk about superfilters, and sets of filters etc.) while subnets (there are different notions) are more cumbersome. Proofs with nets look nicer to people from analysis or algebra, while proofs with filters are more natural coming from set theory. I'd say just learn filters. $\endgroup$ – Henno Brandsma Sep 22 '18 at 6:18

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.

  • $\begingroup$ Thanks. I think it is a good advice. $\endgroup$ – Dog_69 Sep 25 '18 at 15:12
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    $\begingroup$ I have been reading some exercises corresponding to Engelking's General topolohy Chapter 1 and they are awesome. Each of them is a very interesting theorem that you have to prove. Thank you very much! $\endgroup$ – Dog_69 Sep 27 '18 at 22:58

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