continuous bijective maps between manifolds If $f$ is a continuous bijective map between two manifolds $M$ with dimension $m$ and $N$ with dimension $n$. Why this map does not necessarily preserve the dimension i.e $m\not = n$? What is a good example about that?

In the case $n=m$ : Any injective continous maps $\mathbb R^n \to \mathbb R^n$ is an homeomorphism on its image. This is the classical invariance of domain theorem. 
Since an immersion is not an embedding in general, my argument I did wrote is not true for $n < m$, as George Elencwajg said in the comments. 
