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I was tyring to read this fact that the connected sum of two disks is homeomorphic to annulus. By intutitive picture, it is obvious, but I wanted to it in a formalistic way. So I was reading many different textbooks, looking up the formal definition of connected sum, surfaces, etc..

The definition of connected sum includes the notion of homeomorphism. So when they prove the above statement formally, do they use various kinds of homeomorphism, and actually constrct a homeomorphism between the connected sum and annulus?

Anyway in general, if I want to study these things in a fully formalized context, what topics should I study? And could you refer me any rigorous textbook?

Thanks!

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You probably won't find this in a textbook. It's more annoying than anything else. Best to be done as a self-imposed exercise if at all. What have you tried? –  Qiaochu Yuan Oct 21 '12 at 7:47
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Maybe easier to see it is a cylinder? The connected sum by definition here is gotten by gluing two annuli along their inner circles. Simply parameterize one by $S^1\times [0,1/2]$ and the other by $S^1\times [1/2,1]$, so that after attaching you get $S^1\times [0,1]$. –  user641 Oct 21 '12 at 9:28
    
@QiaochuYuan I havne't really tried anything because I couldn't do anything much. (And I'm very new to this topic.) I was looking at all the algebraic textbooks but they only demonstrate this by pictorial arguments. So I was wondering if there is any formalized proof of that. –  julypraise Oct 21 '12 at 9:51
    
@SteveD Thanks for the insight. I will kepp in mind what you said. But I don't think I can do what you've instructed at this stage; I've met this area only recently. I was just wondring if they actually formally prove this and if they do, they are in some textbooks. –  julypraise Oct 21 '12 at 9:54
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This is one of those things that's pretty obvious, but it's a bit of a hassle to actually construct anything more than a rought hand waving "proof" of it. I'd recommend looking at Allen Hatcher's textbook, which can be found here on his website: math.cornell.edu/~hatcher/AT/ATchapters.html. The methods he covers are suited to showing that spaces are or are not homotopy equivalent, or that they aren't homeomorphic, but in my experience it seems to be the standard text for an introduction to algebraic topology –  user123123 Oct 21 '12 at 12:05

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