# Textbooks for an independent study in point-set topology

I am planning to sign up for an undergraduate "course" in point-set topology next semester. It is really an "independent study" in that this course will not have any lectures. It will just have two tests and a final exam. I plan to follow a YouTube playlist on Topology to solidify my understanding of the subject.

Since it is an "independent study," the instructor of this independent study has asked me if I preferred to read a certain topology textbook. So, I researched the books on Topology and decided upon Topology: A first Course by Munkres. However, last semester I had done a similar "independent study" in Abstract Algebra for which I had studied A First Course in Abstract Algebra by Fraleigh which is highly recommended online. However, I found that for an "independent study," this was was rather stingy with examples, although this might just be me. I was still getting acquainted with axiomatic math courses.

Do you believe that the book by Munkres would be a good choice for an "independent study" in Topology? Is it generally skimpy on examples like Fraleigh's book? Would you recommend some other book on Topology? P.S. I would hate to go for a book that is either less challenging or less mathematically rigorous that Mukres's.

Potentially relevant information: So far, the axiomatic courses I have completed include Real Analysis (based on first six chapters from Rudin), Mathematical Statistics (based on Hogg, Craig's first seven chapters), Abstract Algebra (based on Fraleigh), and a course in Complex Variables (based on Churchill's first four chapters). Any suggestions and/or advice are deeply appreciated!

Edit: I am planning to pursue a doctoral degree in economics for which I have been advised that I should take a course in Topology.

• less rigorous that Mukres's --- This needs to be explained further, because there is a mathematical usage of the term "rigorous" (that doesn't really make sense here, since pretty much any text at the advanced undergraduate level or higher is going to be rigorous by any reasonable standard) and there is a natural language usage (roughly, how "hard" or "advanced" the book is). – Dave L Renfro Apr 18 at 9:32
• @DaveLRenfro I've tried to clarify what I meant by less rigorous that Mukres's – Ricky_Nelson Apr 18 at 15:07
• My recollection (from prehistoric times when I was learning this) is that the topology part of "Introduction to Topology and Modern Analysis" by George Simmons was enjoyable to read and covered the parts of general topology that are most useful in other areas of math. (The analysis part of the book is also good, but irrelevant to the present question.) – Andreas Blass Apr 18 at 15:31
• Incidentally, for economics a more analysis oriented approach is definitely preferred over a book whose purpose is to lead you into algebraic topology, and thus the Simmons book becomes even more attractive for you, I think. Maybe read through that (most of the book), with the goal of fully understanding nearly everything, and also have on hand Infinite Dimensional Analysis: A Hitchhiker's Guide by Charalambos D. Aliprantis and Kim C. Border, which was specifically written for economics and is also superbly written (but very expensive). – Dave L Renfro Apr 18 at 17:23
• The Aliprantis/Border book would be useful for graduate work, at least if you wish to pursue economics from a highly mathematical perspective. Possibly also worth having is Topological Spaces: Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity by Claude Berge, since the Dover reprint is cheap and also intended for economics, but I definitely wouldn't recommend trying to use this for your independent study (e.g. p. 19 introduces $\limsup$ and $\liminf$ relative to a filter base for an arbitrary indexed "sequence" of sets). – Dave L Renfro Apr 18 at 17:29

Munkrese is a good book for introductory Topology in part because it has a lot of background at the start. If I recall correctly, the first 70 pages aren't so much Topology as they are just basic abstract math background. You can get a copy pretty cheaply and when you ask questions about it there are lots of people who have also studied it and can help you if you ask.

Fraleigh is a good book, but I think you'd benefit from having a copy of Dummit and Foote's 3rd edition to look at when questions about algebra arise. Similarly, it's good to supplement Munkrese with something else. I like Topology by Marco Manetti, for example see here, but there are dozens of good texts written in the past 20 or so years. Topology, much like Complex Analysis, is a topic which many Mathematicians love and as such find time to write their own book about.

Topology is likely to play a larger role in economics in the future as big data plays a larger role in the function of corporations and governments. Topological Data Analysis(TDA) requires all that fancy schmancy math that is sometimes poo-poo'd around here. TDA is very modern and still very much under development. Looking for large scale patterns in data is a problem which requires the full sophsitication of modern topology as well as a good grounding in mathematical statistics.

Topology is also more and more important to the design of semiconductors from what I've heard from those who know much more than me...

• I also like Gamelin's Topology book since it genuinely tries to merge analysis and topology and it covers a lot of ground. I think it is terse in places though, well, they all are since Topology has intrinsic difficulty... – James S. Cook Apr 18 at 20:33

If you're going to be an economist, even doing highly theoretical work, then the vast majority of the field of topology is going to be of zero relevance to you. Very little of it has any practical application in the sciences. What we actually use in the sciences is mostly just the topology of manifolds, which is a tiny, tiny piece of the larger picture of topology. For a sample of the kind of thing that is studied in topology but that is almost certain not to help you as an economist, see https://en.wikipedia.org/wiki/Long_line_(topology)

So if you just think topology would be a cool, fun thing to learn about, that's fine, have fun! But given that you're an undergrad econ major, you could also choose to focus more narrowly. (I assume the structure of this course allows you to have some say in what topics you learn.) A really fun book with a narrower focus is Topology Now!, by Messer and Straffin.

I am planning to pursue a doctoral degree in economics for which I have been advised that I should take a course in Topology.

I'm extremely skeptical of this claim. It sounds like you just like math, so if you do it, do it for that reason.

• Maybe the John Nash route is what they have in mind? – Peter Saveliev Apr 18 at 19:52
• @PeterSaveliev: Do you know of any examples where John Nash used anything outside of the topology of manifolds? E.g., the Nash embedding theorem is about manifolds. – Ben Crowell Apr 18 at 20:00
• @ Ben Crowell He used algebraic topology for his equilibrium. Not manifolds. – Peter Saveliev Apr 18 at 20:05
• Interestingly, I have small degrees of separation with at least two things you mentioned. One is that a former professor I had 2 topology classes from has written a paper showing there are $2^{\omega_1}$ pairwise non-diffeomorphic $C^{\infty}$ structures on the long line, and the other is that a few years before the book "Topology Now!" was published, I spent a couple of hours looking at an early version of the manuscript with Messer in his office, the evening of a job interview (I was not offered the job, however). – Dave L Renfro Apr 18 at 22:49