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In a lecture we were trying to show a torus $T^{2}_{a,b}$ is homeomorphic to the planar model $I/\thicksim = \left\{ (x,y)| 0 \le x \le 1, 0 \le y \le 1 \right\} $

I need to show,

$\overline{f}([(x,y)]) = (a+b\cos(u))\left(\cos(v)\mathbf{i}+\sin(v)\mathbf{j}\right)+b\sin(u)\mathbf{k}$

is a bijective function. Does it suffice for me to find an inverse function to show bijectivity?

Ok in the lecture to show something is homeomorphic, we needed to show several things but one thing that confused me is how to show bijectivity. Now I know what it means for something to be bijective, but as for showing this, that's another question.

As for recognising answers, how do I do this properly?

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    No. 'functional analysis' and 'functions' are not the tags you're looking for.2011-08-16
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    You haven't accepted answers on most of your questions. Please take into account that this feature exists not only to acknowledge the efforts of those who answered, but also to mark the questions as satisfactorily dealt with so people don't unnecessarily go back to them.2011-08-16
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    Does anyone else find it "fishy" that the asker uses the word "homeomorphic" and then goes on to state that they (only?) need to show that some map is a bijection? You would expect anyone who knows enough mathematics to know of the term homeomorphism would also know that bijectivity by itself is not enough.2011-08-16
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    @kahen, I don't know whether you have ever taught topology. I have, and I'm not the least bit surprised to see students using words like "homeomorphic" whilst not understanding their meaning. Once, on a test, I described two spaces, and asked whether they were homeomorphic. One student answered, "the first one is, but the second one isn't." Go figure.2011-08-16
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    @kahen: It might be meant as "as part of this, I need to show". Quite apart from this particular post, here's a meta thread related to your thought: http://meta.math.stackexchange.com/questions/2606/discrepancies-between-skills-and-ambitions2011-08-16
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    @Gerry: This reminds me of a story my friend likes to tell of someone in his class who asked why a quadrilateral necessarily has four corners. The teacher asked her to draw one that doesn't, she went up to the blackboard, thought for a while, said "ah, I see" and sat down again. I wish I knew what went through her mind...2011-08-16
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    @Gerry: Isn't that the story that was not isomorphic, in contrast to mine that was, from the comments on that long MO thread? :-)2011-08-16
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    Do you know how to show that this is a homeomorphism, and not just a continuous bijection?2011-08-16
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    @Asaf, if I've changed/misrepresented/stolen something from an MO thread, I plead not guilty by reason of comedic license. A good joke justifies many sins.2011-08-17
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    @Gerry: I thoroughly agree about that. Either way, your story is not isomorphic, therefore not homeomorphic either! :-)2011-08-17

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