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Before, I scoured stack exchange for the prerequisites to read this book:

(See: Topology Prerequisites for Algebraic Topology, Module theory for chapters 1-3 of Hatcher Algebraic Topology, Learning Roadmap for Algebraic Topology, Algebra prerequisites for Hatcher's Algebraic Topology)

I got 100 on my point set topology course and am very comfortable with groups, rings, and modules so I thought I should be able to comfortably start learning out of Hatcher. I've heard great things about this book - how it's by far the most readable book in introductory algebraic topology, beautiful typesetting, builds up from basics, assumes little of the reader, etc.

But a couple pages in and I'm completely lost!

Hatcher keeps talking about orientable surfaces and genus - both concepts which are not once mentioned anywhere in Munkres, or indeed, in the majority of introductory topology courses. Neither of these concepts are defined in Hatcher either. So I thought - maybe I'll learn these concepts first and come back to the book! But no - every reference on genus I've found is another book on algebraic topology, one that needs prereqs beyond the ones demanded by Hatcher. All mentions of orientable surfaces lead me to differential geometry references - but I know 0 things about differential geometry.

The kicker is the prof I talked to who is running the course based on this book said I'll be fine in the course given my background knowledge. But from what I've skimmed from the book, I'm sure to fail the course at this rate.

I'll give a different example. The first time Hatcher defines the real projective space is here:

enter image description here

But is it just me or is this 'proof' extremely unrigorous and handwavy? I feel like half this proof relies on the reader's intuition about $\mathbb{R}^n$ for $n<=3$ and somehow this translates to general $\mathbb{R}^n$. The other half is massive leaps in logic that took me a long time just to think of a proof sketch. And this was just one example from the book, mind you.

Honestly, I'm thinking I've just hit a wall in my limits in pure math. What am I doing wrong? What makes this text readable to others but not me?

Or rather, my question should be - how do new people read this book in such a way that they don't get hung up in the way I am doing right now?

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    $\begingroup$ I'm thinking I've just hit a wall in my limits in pure math --- Your comment about being "unrigorous and handwavy" is EXACTLY how I felt with much of algebraic topology, and after some very unpleasant experiences I simply stayed away from it. Mathematics is vast and is practiced in many different ways. I originally liked algebra and point-set topology a lot (late 1970s), so it made sense that I'd like algebraic topology. NO, I didn't. I grew up very interested in physics, especially relativity and the idea of higher dimensions, so it made (continued) $\endgroup$ – Dave L. Renfro Dec 31 '19 at 9:51
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    $\begingroup$ sense that I'd like differential geometry and general relativity. NO, I didn't. By the very early 1980s, I realized I liked rigorous reasoning and analysis (real and functional, especially) and philosophy and (still) physics, so it made sense that I'd like the mathematical and philosophical foundations of quantum mechanics. No, I didn't (this was 2 years of graduate study in early 1980s). After some more flirtations (ordinal number and ordered set arithmetic issues, general and set-theoretic topology, etc.) I finally found some things that actually worked for me. $\endgroup$ – Dave L. Renfro Dec 31 '19 at 9:58
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    $\begingroup$ I think that the references to genus and orientable surfaces are, in the beginning, not to be taken seriously : they're remarks for your future self who will understand those concepts. Many textbooks have this: they talk about advanced stuff very early on, without introducing the relevant concepts/proofs, relying on some intuition they hope their reader has; or hoping the reader who doesn't know a thing about them will just skim through them. As for the example, it's not a "non rigorous proof" : it's simply not a proof ! (cont.) $\endgroup$ – Maxime Ramzi Dec 31 '19 at 10:21
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    $\begingroup$ Regarding your concerns about Example 0.4, I must ask: how are you on quotient spaces? In my experience, I've found quotient spaces to be a real weak spot in many students' preparation for algebraic topology. Every step of the constructions in Example 0.4 can be justified by theorems on quotient spaces from Munkres, and it would be a fantastic exercise for you to work through those justifications. $\endgroup$ – Lee Mosher Dec 31 '19 at 17:45
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    $\begingroup$ I'm not particularly referring to the universal property. In Example 0.4, there are, like five or so different quotient space steps following quickly one after the other: first "$\mathbb RP^n$ is topologized..."; then "$\mathbb RP^n$ is also..."; then "This is equivalent to saying..."; then "... we see that $\mathbb RP^n$ is obtained from...". So when I ask "How are you on quotient spaces?", what I'm referring to is whether you can follow these deductions. $\endgroup$ – Lee Mosher Dec 31 '19 at 23:14

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