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.
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$\begingroup$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.
$\endgroup$ 1 $\begingroup$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.
$\endgroup$ $\begingroup$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.
$\endgroup$ $\begingroup$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.
$\endgroup$ 7 $\begingroup$I think you should also consider Prof. Ronald Brown's book:
Topology and Groupoids
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
$\endgroup$ 2 $\begingroup$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.
$\endgroup$ $\begingroup$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.
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