# Topology Prerequisites for Algebraic Topology

Note: There is another question of the same title, but it is different and asks for group theory prerequisites in algebraic topology, while i want the topology prerequisites.

I am a physics undergrad, and I wish to take up a course on Introduction to Algebraic Topology for the next sem, which basically teaches the first two chapters of Hatcher, on Fundamental Group and Homology. However, I don't have a formal mathematics background in point-set topology, and I don't have enough time to go though whole books such as Munkres. So What part of point set topology from Munkres is actually used in the first two chapters of Hatcher?

More importantly, I wanted to know if the first chapter of the book Topology, Geometry and Gauge Fields by Naber or first 2 chapters of Lee's Topological Manifolds would be sufficient to provide me the necessary background for Hatcher.

• What Sigur wrote in his answer, but also separation axioms, though most spaces you deal with in algebraic topology have all separation properties. Quotient spaces and quotient maps are of particular importance in homology. Feb 12, 2013 at 19:36
• You might be interested in A. H. Wallace, An Introduction to Algebraic Topology, Dover, 2007. It is a reprint of the 1957 original, so it is on the older side and a bit "old fashioned" in terms of the exposition. That said, it assumes no prior background in topology and so builds up the necessary prerequisites in the first three chapters (about 60 pages). Feb 17, 2013 at 3:03
• Other than a little "mathematical maturity" there's not very many hard formal prerequisites for studying from Hatcher. On the point-set topology front, you'll want to be familiar with the subspace topology and the quotient topology. You should also be familiar with abelian groups and at least be modestly familiar with abstract (non-abelian) groups up to quotient groups. Feb 18, 2013 at 5:12

Chapter 1 of Hatcher corresponds to chapter 9 of Munkres. These topology video lectures (syllabus here) do chapters 2, 3 & 4 (topological space in terms of open sets, relating this to neighbourhoods, closed sets, limit points, interior, exterior, closure, boundary, denseness, base, subbase, constructions [subspace, product space, quotient space], continuity, connectedness, compactness, metric spaces, countability & separation) of Munkres before going on to do 9 straight away so you could take this as a guide to what you need to know from Munkres before doing Hatcher, however if you actually look at the subject you'll see chapter 4 of Munkres (questions of countability, separability, regularity & normality of spaces etc...) don't really appear in Hatcher apart from things on Hausdorff spaces which appear only as part of some exercises or in a few concepts tied up with manifolds (in other words, these concepts may be being implicitly assumed). Thus basing our judgement off of this we see that the first chapter of Naber is sufficient on these grounds... However you'd need the first 4 chapters of Lee's book to get this material in, & then skip to chapter 7 (with 5 & 6 of Lee relating to chapter 2 of Hatcher).

There's a crazy amount of abstract algebra involved in this subject (an introduction to which you'll find after lecture 25 in here) so I'd be equally worried about that if I didn't know much algebra.

These video lectures (syllabus here) follow Hatcher & I found the very little I've seen useful mainly for the motivation the guy gives. If you download the files & use a program like IrfanView to view the pictures as you watch the video on vlc player or whatever it's much more bearable since you can freeze the position of the screen on the board as you scroll through 200 + pictures.

I wouldn't recommend you treat point set topology as something one could just rush through, I did & suffered very badly for it...

• Hey, Thanks for the comprehensive answer. I am doing a full masters level course of groups and rings, so I am pretty sure, I will have the algebra prerequisites. I will try to finish Munkres, else I will go through Naber or Lee.
– user23238
Feb 18, 2013 at 8:52
• No worries, just wondering whether you read the introductory chapter of Naber? I found the dirac string stuff fascinating, & the end of the book contains stuff on Donaldson theory that (I think) one of our lecturers contributed to or has some relationship with at any rate. If these books are too brief books like the schaums one or topology without tears are useful for testing definitions out on finite topologies. Feb 18, 2013 at 17:12
• I did read part of Nabers first chapter including the Dirac monopoles, it is very interesting, I agree! I also want to read the section of Hopf Bundle, but not finding the time to do so. Have you read bout the hopf bundle?P.S: +1 for your answer. I will wait for a few days, before I award you the bounty.
– user23238
Feb 19, 2013 at 0:16
• Yeah, & the mention of how Hopf's work apparently had no relation to Dirac strings reminded me of a moment in this talk by Steven Weinberg where he talks about how he cracked a problem with Lagrangians using abstract algebra :p Good introductory chapter to a book... Feb 19, 2013 at 1:37

For sure you'll need continuous functions, homeomorphisms, connectedness, compactness, coverings and many others.

I prefer Munkres over all topology books.

You might starting with Munkres chapter 2, then read chapters 3, 4, 7 (without " * " sections), but if you have enought time is not bad idea reading all of the first part: Chapters 1-8 (long but fun).

I think that chapter 1 is good for you, is an intuitive approach for set-theory, since you are a physicist probably not like going too deeply into sets, but if you dont have time, skip it.

But my biggest advice is not worry about taking the course as quickly, if you don't feel safe. I was physicist.