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Where can I find the Theorem of invariance of the dimension with diffeomorphisms? And about Between Homeomorphism?

I also want to know about the Hausdorff dimension invariance what is deeper!

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What do you exactly mean by invariance of dimension? – azarel Mar 18 '12 at 21:46
Well we have diffeomorphism between two open sets of $R^n$ and $R^m$ then we must have $n=m$. The same for homeomorphism! About Hausdorff dimension it is more subtle! – checkmath Mar 18 '12 at 21:49
Do you want proof? Or what? – Blah Mar 18 '12 at 21:53
Well, I want a reference for the result with a proof would be nice! – checkmath Mar 18 '12 at 21:55

If $f$ is a diffeomorphism between two open subsets of $\mathbb R^n$ and $\mathbb R^m$, respectively. Then $df_x: \mathbb R^n\to \mathbb R^m$ is a linear isomorphism between $\mathbb R^n$ and $\mathbb R^m$, hence $m=n$. If $f$ is only assumed to be a homeomorphism the result is more subtle and you can find the proof in any algebraic topology book (e.g. Hatcher, Rotman, ...)

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About the Hausdorff measures dimensions? – checkmath Mar 18 '12 at 22:22
@chessmath For the Hausdorff dimensions you can find the result in Folland (Real Analysis). – azarel Mar 18 '12 at 22:38

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