Reddit reviews Introduction to Topological Manifolds (Graduate Texts in Mathematics)

Lee is still the easiest and best for self study. https://www.amazon.com/Introduction-Topological-Manifolds-Graduate-Mathematics/dp/1441979395/ref=sr_1_3?keywords=lee+manifolds&qid=1558265795&s=gateway&sr=8-3

followed by:

https://www.amazon.com/Introduction-Smooth-Manifolds-Graduate-Mathematics/dp/1441999817/ref=sr_1_2?keywords=lee+manifolds&qid=1558265795&s=gateway&sr=8-2

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It's long, taking almost two volumes to get to Stokes theorem. If conciseness is important, you can just read Warner:

https://www.amazon.com/Foundations-Differentiable-Manifolds-Graduate-Mathematics/dp/0387908943/ref=sr_1_1?keywords=Warner+manifolds&qid=1558265966&s=gateway&sr=8-1

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Tu looks good, but I haven't read it carefully.

Hey I'm a physics BSc turned mathematician.

I would suggest starting with topology and functional analysis. Functional analysis is the foundation of quantum mechanics, and topology is necessary to properly understand manifolds, which are the foundation of relativity.

I would suggest Kreyszig for functional analysis. It's probably the most gentle functional analysis book out there.

For topology, I would suggest John Lee. This topology text is unique because it teaches general topology with a view towards manifolds. This makes it ideal for a physicist. If you want to know about Lie algebras and Lie groups, the sequel to this text discusses them.

Baby Rudin only treats metric spaces. This is not nearly a general enough background to attach an algebraic topology text.

I would recommend spending more time learning point-set topology first. There are a couple of different ways to do this. One is read one of the books I listed above, or one like it. The other option is to study something else that will teach you the topology along the way. This is kind of specific, so I'll just say the book I'm thinking of is Topological Manifolds by Lee. Chapters 2-4 will cover an introductory topology class. Chapter 5 will teach you about CW complexes, an extremely important topological construct. The rest of the book actually dives into some algebraic topology, so if you stick with it, you'll get the AT intro you were looking for.

Of course, once you've mastered the basics, the classic text for AT is Hatcher (free PDF). I still find it to be a challenging read, but it definitely lays the groundwork for further study in topology.

I love Jack Lee's series on manifolds:

Introduction to Topological Manifolds

Introduction to Smooth Manifolds

I've heard Munkres' Topology is fantastic as an introduction to general topology, but never read it myself.

Study what you find the most interesting!

Does your linear algebra include the spectral theorem or Jordan canonical form? IMHO, a pure math subject that is relatively the easiest to learn and is useful no matter what you do is linear algebra.

Group theory (representation theory) has also served me well so far.

If you want to learn GR and Hamiltonian mechanics in-depth, learning smooth manifolds would be a must. Smooth manifolds are basically spaces that locally look like Euclidean spaces and we can do calculus on. GR is on a pseudo-Riemannian manifold with changing metric (because of massive stuffs). Hamiltonian mechanics is on a cotangent bundle, which is a symplectic manifold (whereas Lagrangian mechanics is on a tangent bundle.) John Lee's book is a gentle starting point.

Edit: If you feel like the review of topology in the appendix is not enough, Lee also wrote a book on topological manifolds.

Set Theory:

Naive Set Theory

Number Theory:

Elementary Number Theory

Introduction to Analytic Number Theory

A Classical Introduction to Modern Number Theory

Topology:

Topology

Introduction to Topological Manifolds

Okay, I'm going to go against the mold here and say Lee's Topological Manifolds is your best bet. This book seemed to be the clearest, most thorough treatment of the topic of the topic at an introductory level. I think it's definitely a lot more verbose than Munkres, but I see it as an advantage as a lot of details are spelled out explicitly (which I something I like to see in textbooks that introduce a new topic to a unfamiliar audience). Plus the extra emphasis on manifolds is a very nice feature of the text.

That is just my recommendation, though. I think Munkres is a great book as well, but Lee's book seemed to present things in a way that clicked with me a bit better.

Armstrong and Lee are both worth checking out.

I prefer terse books for some reason, so Armstrong is my personal favorite, but other people may prefer Lee, which includes lots and lots of explanations and examples.

Though not an intro to topology book in the strictest sense (it doesn't go into too much detail on seperation axioms, metrization theorems, or Tychnonoff's theorem) I found going through Lee's Introduction to Topological Manifolds to be a fantastic book to learn from, especially if you're interested in going into differential geometry afterwards. I find that many a topology book commits the sin of focusing way too much on curious pathological counterexamples and the beginning students find themselves awash in a sea of formalities without any intuition to guide them. Having many geometrically motivated examples will make the subject highly approachable and eases the transition into thinking about abstract topological spaces.