The definition of a manifold in my course notes is given as follows:
Let $\mathcal{M}\subseteq\mathbb{R}^n$. A smooth chart on $\mathcal{M}$ consists of a subset $U\subseteq{\mathcal{M}}$, open in $\mathcal{M}$, an open set, $V\subseteq\mathbb{R}^m$, and a diffeomorphism $\phi:U\rightarrow{V}$. A smooth atlas for $\mathcal{M}$ is a collection of charts whose union covers $\mathcal{M}$. We say that $\mathcal{M}$ is a smooth manifold of dimension $m$ (or a smooth $m$-manifold) if it admits a smooth atlas with charts mapping to $\mathbb{R}^m$.
My problem here is that we have a diffeomorphism mapping an open subset $U$ of $\mathbb{R}^n$ to an open set $V$ of $\mathbb{R}^m$. This is then necessarily a homeomorphism, contradicting invariance of dimension (unless $m=n$, but my notes simply say that it is 'easily seen that $m\leq{n}$').
Is there an issue here or am I missing something?
${\cal M}$ is contained in $R^n$ and not always an open subset subset of $R^n$. think of $R^2$, ${\cal M}=R\times\{0\}$, you have a diffeomorphism between ${\cal M}$ and R