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I'm an undergraduate math student. I've taken several courses in topology yet I always feel like I'm at the most basic level of knowledge there is (I think of it as a natural number - it can be big and get bigger as much as I want, but it will never even grasp infinity). Introductory books such as Willard's, Dugundji's, Engelking's and Munkres' really are veeery elementary, but I can't seem to find something more advanced anywhere.

There is differential topology, continuum theory, knot theory, algebraic topology (and all the abstraction that become the theories of homology and homotopy) and the theory of topological vector spaces. I've come across some concepts like door spaces, fuzzy topological spaces, hyperspaces, topological data analysis and persistent homology, the connectification of spaces (analogous to compactifications) and even the strange concept of the "tight span" of a metric space. We have the newborn hyperspace of non-trivial sequences, and I've also read about toposes and the categorical notion of a closure operator.

There are also websites like Dan Ma's Topology Blog, $\pi$ Base (which is a web-version of Counterexamples in Topology) and this strange site with a ridiculous amount of examples in continuum theory.

It seems like a vast world of topology is just developing and evolving, but it seems quite inaccesible. I'd like to learn a lot more than the very few things I mentioned, because I know there is a lot more out there, however I simply don't know how to do it.

How do I start? Where do I go next?

PatrickR
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R Los
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  • Yea, I'm in a similar boat to you. I think alg top is prob the next best step. There's a whole lifetime out there. – Rushabh Mehta Nov 26 '19 at 07:22
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    Patience, young padawan. Your question is a little ridiculous: you’ve given a laundry list of more or less totally unrelated ideas, stitched together only by the generic thread of having something to do with the notion of a topology. The short answer is, talk to your advisor about the kind of math you like and what to study next. I’ve voted to close, because this site isn’t for personal advice. – symplectomorphic Nov 26 '19 at 07:28
  • @symplectomorphic I'm sorry. I just find so much joy in topology that I'm a little desperate to learn more every day... – R Los Nov 26 '19 at 07:31
  • No need to apologize, and most of us on this site find joy in math. But we don’t know enough about you — what your professional goals are, what other courses you’ve already taken, etc. — to provide useful advice. That’s why this isn’t a site for personal advice; you should talk to your advisor and your professors. – symplectomorphic Nov 26 '19 at 07:36
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    Engelking (General Topology, 2nd ed.) is not that elementary. Have you tried doing all the exercises ? – Henno Brandsma Nov 26 '19 at 08:09
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    You might want to check out the survey books "the Handbook of Set-theoretic topology", "Recent Progress in Topology" (3 vols now), "Open Problems in Topology" (2 vols.). Check your library ! – Henno Brandsma Nov 26 '19 at 08:11
  • Possible duplicate of Where does a Topology student go after Munkres? – Moishe Kohan Nov 26 '19 at 15:27

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I think at the point you are topology splits into various distinct fields that have little to with each other and can have very different prerequisites. Some big areas ordered by how topological I think they are:

  • point set topology: most 'pure' topology, only requires some basic set theory
  • metric geometry: requires some analysis, differential geometry can be very helpful for intuition and understanding the origin of ideas
  • differential topology: requires analysis and then differential geometry
  • homology and cohomology: requires abstract algebra
  • algebraic topology: requires lots of abstract algebra, least geometric/ topological
quarague
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