Where does a Topology student go after Munkres?

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.


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).

Tags:

Reference Request

General Topology

Soft Question

Differential Topology

Algebraic Topology

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