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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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    Well, I want a reference for the result with a proof would be nice!2012-03-18

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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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    @chessmath For the Hausdorff dimensions you can find the result in Folland (Real Analysis).2012-03-18