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I was reading Baby Rudin for Real Analysis and wanted to explore Topology a little deeper. I bought George Simmons' Introduction to Topology and Modern Analysis and found myself liking it. I am having some problems every once in a while with prerequisites.

How much Set theory do I need to learn before diving into the aforementioned book?

Also, the book is divided into 3 parts : Topology, Operators and Algebras of Operators.

Till where can I trot with a good understanding of SV Calculus?

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I suspect you need no more than what you might learn in a first semester of real analysis. –  mixedmath Jun 25 '12 at 21:30
    
To start general topology you won't need much set theory at all, but once you get deep enough, it's basically the same as advanced set theory... –  Zhen Lin Jun 25 '12 at 21:30
    
@ZhenLin, I don't know anything beyond Unions, Intersections, Venn diagrams and a few basic theorems. Sufficient? –  Real_Analysis Jun 25 '12 at 21:31
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From my experiences with the Simmons book, which I thought it was a wonderful introduction to the subject(s) - you don't need any prerequisites in set theory at all, as he covers all the set theory he needs in the first few sections of the book. Just stick with Simmons for now, and don't bother with any other books. –  Old John Jun 25 '12 at 21:37
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You should need essentially no set theory to start in on topology. Your current knowledge seems fine. –  KReiser Jun 25 '12 at 21:45

2 Answers 2

up vote 4 down vote accepted

Simmons' book has a brief introduction to set theory at the beginning. If you can get through it without any trouble, you probably have a strong enough background in set theory. In the worst case, there may be a couple of things that you will get stuck at. If that happens, you can go try to figure it out on your own, and if that fails go back to a reference for set theory. This should be pretty infrequent.

The best thing to do is to try to read the book (and understand every line of every proof, not just skimming over it). If you get stuck, find a way to get past it. If you find yourself frequently getting stuck, figure out what it is you're weak on, and go study that before you go back to the book.

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My recommendation would be: carry Simmons book with you wherever you go, read it, master it, solve every problem in the exercises he gives (the are mostly very good) - and when you get really stuck, ask for help on math.stackexchange ... –  Old John Jun 25 '12 at 21:47

I assume that in the book you are asking about the preliminaries included by author would be sufficient.

One thing which I consider useful is Zorn's lemma, which is in the first chapter of Simmons' book. But even if you choose different text, you will bump into using Zorn's lemma a few times.

Other than that, the only thing that I can think of are ordinals and transfinite induction. But they are perhaps less important. Many of the books that are intended as the first course in general topology don't include them. (For instance, Willard includes these two topics.)

However, I don't think you should worry too much about preliminaries. You can always get back to some topic, if you find at some point in the book, that you need to know something from set theory (or any other area).

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