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Let $X$, $Y$ be topological spaces and assume there exists a continous bijection $X\to Y$ and there exists a continous bijection $Y\to X$. Are $X$ and $Y$ homeomorphic?

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    That sounds like a great homework problem! Any thoughts on why it might *not* imply that the spaces are homeomorphic? What's missing? Let us know what you've tried, and we can better help you. – 2017-02-21
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    @JohnHughes Well, obviously a continous bijection does not have to be a homeomorphism. There are plenty examples, for example there is a continous bijection $[0, 1)\to S^1$ even though they are not homeomorphic. However in the example I've shown there is no continous bijection $S^1\to [0, 1)$. So I just thought about this question. I have no idea how to attack it though. It doesn't seem trivial at all, I wouldn't call it a homework. – 2017-02-21
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    I, for one, don't necessarily think this is a homework problem, nor do I know the answer. It's an interesting question, though, because the usual "each of $[0,1]$ and $(0,1)$ can be embedded in the other" doesn't seem to help. – 2017-02-21
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    Well..you supposed not that you have a continuous bijection, but that you have *two*, one in each direction. That's just about the setup for "homeomorphism" except that one thing is missing. If I had to guess, I'd figure that the exception was something that involved weird topologies; if everything is a nice compact manifold, then it might be the case that $X$ and $Y$ *are* homeomorphic (but I wouldn't bet it on it just yet). – 2017-02-21
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    @JohnHughes Being compact is obvious, since any continous bijection from a compact space has to be closed (thus a homeomorphism). So I'm rather looking at some noncompact and probably pathological cases. – 2017-02-21
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    See http://mathoverflow.net/questions/30661/non-homeomorphic-spaces-that-have-continuous-bijections-between-them for a pretty complete answer. (Indeed, it showed up in a link-chase from the "Related" links to the right of your question, namely http://math.stackexchange.com/questions/1125293/two-non-homeomorphic-spaces-with-continuos-bijective-functions-in-both-direction?rq=1). – 2017-02-21

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