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Studying Topology for Physics
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I would recommend Introduction to Topology: Pure and Applied by Colin Adams and Robert Franzosa.

It's an underrated book imo. It has lots of pictures and examples, and almost every chapter has an application section at the end. You'll find all the essential point set Topology and some elementary algebraic topology in this book. Unfortunately the section on manifolds isn't the best imo but it's great as an introduction. I wish I'd known about this book when I first started learning topology.
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I would recommend looking at a topology chapter on metric space topology in a real analysis textbook and then start working through Munkres or Lee.
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I say dive straight into

Matt Visser

Lorentzian Wormholes: From Einstein to Hawking (AIP Series in Computational and Applied Mathematical Physics)

Use Wikipedia to Google what you don’t know

If I remember correctly the first few chapters have some really good background on the topics you’ll need. You’ll enjoy how applied and cool the later chapters are

(Added a few sentences and edits)
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You can study the first 2-3 chapters of Lee's Intro to Topological manifolds and then jump into his Smooth manifolds book easily.
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You don’t need much point set topology.  First three chapters of  
Armstrong: *Basic Topology*  
should be enough.

As a physicist you may be interested in  
Gregory Naber: *Topology Geometry and Gauge Fields*

The author explains the needed topology when its needed
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Well, first things first. For the physics understanding of manifolds, you don't need to know everything about topological manifolds.

It is sufficient to just look at an introductory course on differential geometry, take a look at Riemannian Geometry and then Lorentzian geometry. That's assuming you want manifolds for relativity purposes.

For Hamiltonian systems it is interedting to learn about symplectic manifolds, but again, for physics purposes you do not need to understand or know every topological property of manifolds. You just need to know what charts are and what differential forms are. But, again, gor a physics point of view you do not need to know why you can define those. You just need to know you can and what they do.

That being said, I can only applaud you if you do want to study topology at a base level such that you know the "why" parts. Personally, my professors always wrote their own lecture books in LaTeX, so I'm not sure which books are good for entry level topology and differential geometry, so I can't attest for the books that others have linked here. I only used more advanced books for my Master Thesis.
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Take a look at the Visual Complex Analysis starting from chapter 7, really good introduction to Topology from the Physics point if view.
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