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Some time back, I tried reading Rudin's Principles of Mathematical Analysis and I found no trouble with the introductory chapter. In chapter II I encountered point set topology. The number of theorems packed into some pages kind of overwhelmed me. I got stuck there as I felt I did not fully understand everything geometrically. I thought topology was supposed to be geometric.

So, I request you to suggest something in this regard. I am contemplating learning some basic point set topology from elsewhere (I don't know where from) and I am wondering if this is the solution to my problem. (Yes, as a preparation for analysis. Even some good notes would be nice or perhaps a great book)

Any help is appreciated.

PS: I hate compromising rigor.

Martin Sleziak
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7 Answers7

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You're right, I think Rudin's Chapter 2 is probably not the best place to first learn point-set topology due to how dense and concise his writing is.

James Munkres' Topology is one of the most common introductions to general topology, and it has some nice pictures in Chapter 2 to give some geometric intuition where topological spaces are first introduced. I like this book. The majority of the exercises are not overly challenging, so it helps to get familiarity with the subject.

I also like Stephen Willards General Topology which is similar to Munkres, but I'd say it's slightly more difficult than Munkres' book.

Finally, although a little older, Kelley's General Topology is a good reference on general/point-set topology, but probably better suited for use after going through some of the previously mentioned books.

yunone
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  • I don't think anyone here can make better recommendations as far as just point-set topology goes. The deeper question is whether or not a full semester course on point set topology is really the best way to learn the subject. But that's a subject for a whole other question............ – Mathemagician1234 Aug 07 '12 at 07:10
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If you are looking for a gentle introduction to basic topology ideas with examples and explanations which give some intuition about what is going on, I would recommend that you could have a look at Simmons book "Introduction to Topology and Modern Analysis".

It is available from Amazon here. Some of the reviews on that page are pretty good, and I would not disagree.

Old John
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The most standard reference for general topology is James Munkres "Topology," though I'm not personally a fan. I'd recommend Klaus Janich's "Topology" as a much more entertaining, well-motivated, and concise read. It lacks exercises, though, and you probably won't fully develop your intuition until you do some.

For this, you can't do better than the Russian school, and the faculty in St. Petersburg has a text they've published here.

Finally, if you find yourself catching the general topology bug, to understand all the fine distinctions between various forms of countability, compactness, and separability, Steen and Seebach's "Counterexamples in Topology" is a classic which hasn't been matched anywhere. Note that the stuff in this last book is not necessary for real analysis, but if you get interested in functional analysis it may well come in handy.

Kevin Carlson
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I *do not*recommend any point-set topology book (such as by Munkres) as a help with Baby Rudin. What I do recommend (and that worked for me) is to get more elementary introduction to Analysis (such as Introduction to Analysis by Kirkwood, which, IMHO, is highly accessible on the undergrad level) and read it through first. You'll get all the topological concepts you need for analysis.

Rudin never meant to be user-friendly and definitely not the book for self-study, but if you still want to read it on your own, there's a "Guide to Baby Rudin" somewhere on ucdavis site (search the web). That might help.

Finally, a point-set topology is highly abstract, closely connected to a set theory (and originaly was thought of as being a part of set theory, like in Hausdorff's Grundzüge der Mengenlehre), the topological concepts used for analysis (such as compactness) has very little geometry in it. I took the course of topology (w/Munkres) only after I went through Rudin.

PPJ
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    "Rudin never meant to be user-friendly and definitely not the book for self-study" That seems like a strong claim: how do you know this? It seems to me that to "use a book for self-study" means, more or less, to read it, and that most books are meant to be read. – Pete L. Clark Aug 07 '12 at 03:38
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    @Pete I'm sorry-I have to agree with PPJ on this. Rudin always seems to me to be more interested in finding the most clever and concise proofs rather then to try and explain the material to beginners. I think there are much more readable but equally challenging real analysis textbooks for talented undergraduates now.My favorite is Charles Chapman Pugh's REAL MATHEMATICIAL ANALYSIS,which is on par with Rudin for sheer difficulty of exercises,yet is much clearer and insightful. This book should also be of interest to the OP since Pugh emphasizes topology more strongly then even Rudin. – Mathemagician1234 Aug 07 '12 at 07:18
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    Rudin won the inaugural Steele Prize for Mathematical Exposition in 1993 and was cited specifically for his Principles (along with another text). To say that the now deceased author of one of the most widely used and respected math texts of the 20th century "never meant to be user-friendly" is a rather extreme accusation. If you mean to say that in your opinion/experience Rudin's Principles is not an easy read and/or you like other texts better, please say that instead, rather than attributing ill motives to Rudin. – Pete L. Clark Aug 07 '12 at 08:34
  • @Pate I meant no disrespect or derogatory statements-I was simply giving my opinion of the book. I'm certain to professional mathematicians-particularly analysts-the books appears incredibly good. Hence the award. What these experts forget-or were such gifted students that they never had this problem-is that this kind of austerity completely devoid of motivation is not only frustratingly obscure to the beginning student,it's also very unpleasant to read-which only deepens their frustration. – Mathemagician1234 Aug 07 '12 at 18:25
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    @Mathemagician1234 : Rudin's book is a classic, and for very good reasons. You shouldn't assume that your experience is shared by anyone but yourself. Please stop making grand pronouncements about what all beginners feel. If you find Rudin's book too difficult, maybe you're just not prepared for it. It's not intended for weaker students. The students at my university (which is a good but not top school) struggle with it, but they manage to learn from it. The hard work they put into reading it serves them in good stead as they continue their studies. – Adam Smith Aug 07 '12 at 18:34
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    @Adam I could have gone the rest of my life without experiencing your condescending commentary again,but such is life. I actually gave the book a conditionally positive review at MAA Reviews Online.I simply think there are less concise and friendlier books that don't spoon feed the students and yet are much more pleasant to learn from. Pugh's book mentioned above is my favorite.Another is Kenneth Hoffman's ANALYSIS IN EUCLIDEAN SPACE,now available in a very cheap Dover edition. I love the exposition in Tom Apostol's MATHEMATICAL ANALYSIS,but unfortunately the exercises are way too soft. – Mathemagician1234 Aug 07 '12 at 19:00
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    It's not a question of liking or not liking the book: both are statements of opinion, hence perfectly valid. My point was that the OP made negative statements about Rudin's motives when writing the book. These are not a matter of opinion, and I find it unseemly (and certainly, unnecessary) to make such statements -- without supporting evidence, I mean -- about a person who is no longer around to confirm or deny them. – Pete L. Clark Aug 08 '12 at 01:23
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I think you should also consider Prof. Ronald Brown's book:

Topology and Groupoids http://pages.bangor.ac.uk/~mas010/topgpds.html

I have personally studied it and even wrote some code in Mathematica to understand its theories.

It is a grand book indeed every student of mathematics should study it

Dara

Dara
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  • +1. An awesome book all students of topology should read PERIOD. It and John Lee's INTRODUCTION TO TOPOLOGICAL MANIFOLDS are the top contenders in my mind to replace Munkres as the standard undergraduate topology books for strong math majors. – Mathemagician1234 Aug 07 '12 at 07:12
  • This book is currently my favorite topology textbook. Love it. –  Aug 11 '12 at 20:33
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The question does mention a bias towards analysis, whereas my own book is biased to geometry and algebraic topology. I liked the book R. P. Boas "A primer of real functions" because of its plentiful examples, and applications of major theorems, such as the Baire Category Theorem, and not at all stodgy.

I also think that nowadays books on topology, especially those biased towards analysis, should do something on Hausdorff distance and fractals, partly because the term fractal is known generally by the public, and so maths students should have some knowledge of the proper mathematical background. I also found that students liked a basic course on it; they could look up books and the internet for information on "the importance of fractals", could download fractal programs, and could do sums on finding specific Hausdorff distances between specified sets in the plane. So it can be made a fun course.

Ronnie Brown
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To get a overall view of topology and its purpose of study, my choice is Topology and Groupoids by R. Brown. Also one can understand why to introduce groupoids instead of groups in Algebraic topology.

Amartya Goswami
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