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!
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, ...)