Where does a Topology student go after Munkres?
Firstly I apologize if this is a bit of a soft question, it's hard for me to ask this quite concretely so I do apologize if this post doesn't seem like I'm asking something immediately.
Let me explain a bit of my background, Topology is the area of mathematics that I enjoy doing the most by far, with regards to the books I've read, I've gone through most of the first 300 pages of Topology: A First Course by Munkres (I still have a few things after Seperation Axioms and stuff to go through). I've also read a bit of Willard's General Topology
I'm currently studying Algebraic Topology and Differential Topology (and Differential Geometry) on my own, and I'm thoroughly enjoying it, but currently it seems that Algebraic Topology and Differential Topology, don't use that much General Topology apart from Compactness, Connectedness and the basics. I've yet to see (in my limited knowledge of Alg and Diff Topology) any real use of things like Separation Axioms and deeper theory from General Topology.
E.g for Differential Topology if your topological space $M$ is second countable and Haursdoff, then you're good to go.
While I certainly have a lot more Differential Topology and Algebraic Topology to learn (and I look forward to it), I also feel like I should learn a bit more of General Topology.
The reason I've given this long explanation (because I hope it will also help others studying Topology who have similarities), is because the path most Topology students follow is the following
$$\text{General Topology} \to \text{Algebraic Topology/Differential Topology}$$
in which one would usually read Topology A First Course by Munkres or a similar intro to general topology book, then follow that with something like Algebraic Topology by Hatcher and Differential Topology by Guillemin and Pollack and Milnors Topology from the Differentiable Viewpoint.
Now (finally) onto my two questions.
If I want to broaden my knowledge of General Topology, what book do I go to next after Munkres? Should I learn some Pointfree Topology (Frame Theory)?. Also I should mention that I don't want to specialize in General Topology.
I hope to someday specialize in Algebraic Topology or Differential Topology/Differential Geometry, so would learning more about General Topology have any direct benefit to my studies of these subjects?
Solution 1:
Let me convert my comment to a full answer:
Unless one is (and you are not!) planning to write a PhD thesis in General Topology, Munkres is (more than) enough.
Depending on what you are planning to study later, you might encounter an issue requiring a bit more General Topology (e.g. proper maps and proper group actions, which you will find in Bourbaki), but you learn this on "need to know" basis (just pick up a General Topology book and look it up when necessary). Instead, my suggestion is to start reading Guillemin and Pollack, and Hatcher (or Massey).
In addition, you would want to (or, rather, have to) learn more functional analysis (say, Stein and Shakarchi) and PDEs (say, Evans) which will be handy if you are planning to go into modern differential topology (which most likely will require you dealing with nonlinear PDEs, believe it or not), and, in case of algebraic topology, - basic category theory (at least be comfortable with the language), Lie theory (at least to know the basic correspondence between Lie groups and Lie algebras), see suggestions here. Yes, General Topology is fun and there are many neat old theorems that you will learn by studying it in more detail, but you have to prioratize: Life is short and your time in graduate school is even shorter.
Solution 2:
For more on general topology you could take selected parts of General Topology by R. Engelking. It includes a huge amount of material in the Exercises and Problems as well, which, if presented in full, would make the book unmanageably big. He also includes bibliographic references in the Exercises and Problems for all the original publications of the deeper ones (and much of the other ones too).