So I am kind of confused by this:
We have $U \subset \mathrm{R}^m$ and $V \subset \mathrm{R}^n$ as open sets. If $f: U \rightarrow V$ is a diffeomorphism then we essentially have $m=n$.
Same holds true if instead of open sets we are working with submanifolds, that is $\mathcal{M}$ and $\mathcal{N}$ are submanifolds of dimension $m$ and $n$ then if $f:\mathcal{M} \rightarrow \mathcal{N}$ is a diffeomorhism then $m=n$
However with arbitrary sets this is apparantly not true. Let $U \subset \mathrm{R}^m$ be an open set and $X \subset \mathrm{R}^n$ be an arbitrary set. Then if $f: U \rightarrow X$ is defined as a diffeomorphism then $m\le n$. My question is can then $f$ even be defined as a diffeomorphism? And if so, can some one comment on why this dimensional invariance fails here and what condition is needed to ensure dimensional invariance.
Thanks
EDIT Maybe an example of a diffeomorphism from an open subset to an arbitrary subset for which the derivative map is injective but not surjective