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This question comes from Conlon's Differentiable Manifolds (it's Exercise 1.1.13).

Let $X$ and $Y$ be connected, locally Euclidean spaces of the same dimension. If $f:X \rightarrow Y$ is bijective and continuous, prove that $f$ is a homeomorphism.


I think I need to use the local homeomorphism $\Rightarrow$ global homeomorphism idea, but I'm having trouble constructing the local homeomorphism. Obviously if I were to do so these local homeomorphisms would have to come from composing the homeos that we have into $\mathbb{R}^n$, but I keep having trouble because we don't know $f$ is a homeo yet. Am I even on the right track?

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    I won't claim this is a full answer, but I think you may want to show that any bijective continuous function between to open sets in euclidean space is an open map, and then this should follow. EDIT: This may be using a flamethrower to get rid of leaves in the lawn, http://en.wikipedia.org/wiki/Invariance_of_domain http://math.stackexchange.com/questions/59532/bijective-continuous-function-on-mathbb-rn-not-homeomorphism2012-04-13
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    @JohnStalfos A bijective continuous function between two open sets in a Euclidean space should be a local homeomorphism because a bijective continuous function from a compact space to a Hausdorff space is a homemorphism. Now a bijective local homeomorphism is a homeomorphism. Doesn't this settle what you state without Invariance of Domain?2015-06-04
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    @caffeinemachine This is not quite right. What prevents the map from taking an open ball to an open interval while somehow still being bijective and continuous?2016-12-27
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    @FanZheng You are right. My reasoning is not proper. Thanks.2016-12-27

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