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Possible Duplicate:
Are continuous self-bijections of connected spaces homeomorphisms?

I know that a function $f\colon X\to Y$ need not be a homeomorphism if it is:

  1. A continuous bijection (e.g. $f:[0,2\pi)\to S^1$),
  2. An open bijection (e.g. inverse of above function),
  3. A bijection between homeomorphic spaces (e.g. $x\mapsto -x$ on $\mathbb R$ with lower limit topology),

but what if $X,Y$ are already known to be homeomorphic, and $f$ is a continuous bijection. Must f then be a homeomorphism, or does there exist a counterexample?

I have given this some thought, but I have so far been unsuccessful at generating either a proof or a counterexample. Also, my searches have turned up unsuccessful.

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    There is a MO thread [Non-homeomorphic spaces that have continuous bijections between them](http://mathoverflow.net/questions/30661/non-homeomorphic-spaces-that-have-continuous-bijections-between-them). I found it in a comment to [this question](http://math.stackexchange.com/questions/20913/are-continuous-self-bijections-of-connected-spaces-homeomorphisms), which was shown between related links on the right.2012-01-12
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    @Martin: That MO thread doesn't seem to have much to do with this question. The MSE question you link does answer this, however.2012-01-12
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    @Nate I am not that sure about closing (as *exact* duplicate). As this question does not require connectedness, perhaps it could be possible to construct simpler counterexamples here, which might help the OP...?2012-01-12
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    @Martin: Perhaps. I voted to close because, of the two best answers I came up with, one turned out to be connected (and to have been given as an answer to the earlier question) and one was essentially that given by the OP in the earlier question itself.2012-01-12
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    Ah, thank you. @Martin, I did find that first link, but I did not manage to find the second. It is a good example for a connected space. For a disconnected space, would it not just work to omit the x-axis?2012-01-12
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    I think it would be hard to beat [Mike's unzipped pants space](http://math.stackexchange.com/a/20932/822) for simplicity, whether you care about connectedness or not. If you really want a *disconnected* example, add an isolated point.2012-01-13

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