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I am planning to take a graduate topology class that uses Algebraic Topology by Hatcher. In order to prepare for that class, would going over chapter 1-8 of Munkres Topology be sufficient enough to prepare for the class? By that time, I would already prepared with the algebra prerequisites.

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    $\begingroup$ Yes, you would then have the prerequisites to read Hatcher (but nonetheless don't necessarily expect that reading Hatcher will be easy). $\endgroup$
    – Qi Zhu
    Jun 10, 2022 at 12:45

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Yes, but you don't need all of it:

  • Chapter 1 covers basic set theory, and it has little to do with topology directly. You'll need it, but you can skip over it if you're already familiar with it.
  • Chapter 2 covers the basic constructions in topology: open and closed sets, continuity, etc. Absolutely necessary. The part on quotient spaces is particularly important, since their topology can be a bit unintuitive (e.g., for $\mathbb{RP}^n$ and general lens spaces), and they recur extensively throughout Hatcher.
  • Chapter 3 covers connectedness and compactness, which is really the meat of point-set topology at this level. Also absolutely necessary.
  • Chapter 4 covers separation axioms. Hatcher works in the CW-complex category throughout (which makes sense for algebraic topology, especially at this level), and CW-complexes are normal. It's important to know some useful results of normality, like the Urysohn lemma and the Tietze extension theorem (and it's important to know what, e.g., the Hausdorff condition is), but it's less important to be familiar with criteria ensuring that spaces are normal, regular, etc.
  • Chapter 5 covers the Tychonoff theorem and compactification constructions. The former is important, but the result itself is less important than its proof (and I personally think that there are better treatments of it than Munkres', which uses an ad hoc version of the ultrafilter approach without actually mentioning the term). The Stone-Cech compactification is useful in general, but I don't think it's that important at this level; the spaces it generates are too badly behaved for most of what Hatcher does. The one-point compactification is simpler and more immediately useful, though.
  • Chapter 6 covers metrization and paracompactness. You can skip it; CW complexes are metrizable with some reasonable conditions on their dimensions, and topological manifolds (let alone smooth manifolds, which is probably the case you're most interested in) are metrizable under some minor technical assumptions. This chapter is a bit outdated now; metrizability conditions aren't as prominent in topology as they used to be, and they're not going to come up in Hatcher. The section on paracompactness may come up (cf. Milnor and Stasheff), but Munkres' treatment is a bit outdated there as well, especially for the applications you have in mind; you might have better luck with a book on manifolds.
  • Chapter 7 is useful for general point-set topology and analysis, but I don't think it'll come up in Hatcher. You can skip it.
  • Chapter 8 covers the basics of $\pi_1$, but Hatcher does it better. Skip it.

In short, I'd suggest you cover chapters 1--3 carefully; be familiar with the material in chapter 4 (especially the Urysohn lemma and Tietze extension theorem, though their proofs can be confusing the first time you run across them); and read the Tychonoff theorem and the one-point compactification sections of Chapter 5. Chapter 0 of Hatcher itself covers a decent bit of topology, and the appendix of Hatcher is a great self-contained introduction to CW complexes; you'll definitely need it in the rest of the book. The first chapter of Spanier's "Algebraic Topology" also covers some material that's useful, although there's a significant overlap with Hatcher. Munkres' text puts an emphasis on codifying exactly what properties of topological spaces are useful for analysis and (in the opposite direction) dealing with very pathological spaces, neither of which is really what Hatcher is about.

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    $\begingroup$ Very nice answer! $\endgroup$
    – Paul Frost
    Jun 10, 2022 at 13:44
  • $\begingroup$ @PaulFrost: Thanks! $\endgroup$
    – anomaly
    Jun 10, 2022 at 13:58
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    $\begingroup$ I would suggest a strong emphasis on the quotient topology. Lack of experience with the quotient topology is, in my experience, a major reason for downfall when reading the early chapters of Hatcher. See my comments to this post for example. $\endgroup$
    – Lee Mosher
    Jun 10, 2022 at 16:06
  • $\begingroup$ @LeeMosher: Thanks, that's a good point! I've updated the post. $\endgroup$
    – anomaly
    Jun 10, 2022 at 16:13
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    $\begingroup$ @user264745: It's a rough one, yeah. The hard part is just finding enough continuous functions on an arbitrary normal space; without a metric, there's no obvious way of generating them. (To that end, you also need some version of the axiom of choice--- not the usual one for arbitrary sets, but stronger than the reasonably unobjectionable axiom of countable choice). Sounds like you're comfortable with it now, but looking at the vastly easier case of a metric space might make the argument more transparent. $\endgroup$
    – anomaly
    Jun 11, 2022 at 14:49
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Hatcher's book begins at the very beginning of topology, topology of sets of points. I think you can simply start with Hatcher's book.

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    $\begingroup$ As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. $\endgroup$
    – Community Bot
    Jun 10, 2022 at 12:27
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    $\begingroup$ This answer is very much incorrect. In Hatcher's book there are no definitions of basic concepts like that of a topology or compactness. From the introduction: "In terms of prerequisites, the present book assumes the reader has some familiarity with the content of the standard undergraduate courses in algebra and point-set topology. In particular, the reader should know about quotient spaces, or identification spaces as they are sometimes called, which are quite important for algebraic topology." $\endgroup$ Jun 10, 2022 at 16:53

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