Bijective immersion is a diffeomorphism

Let $\psi:M\longrightarrow N$ be $C^\infty$, bijective immersion, the $\psi$ is a diffeomorphism. I am having trouble in proving this statement.

What I have done so far is this : By inverse function theorem, it is enough to prove $d\psi:M_m\longrightarrow N_{\psi(m)}$ is an isomorphism $\forall\ m\in M$. We already have that $d\psi$ is injective for all $m$. So it is enough to prove that $d\psi$ is surjective $\forall\ m\in M$. Suppose there is a point in $M$ where $d\psi$ is not surjective, then this means that $dim\ M=p<d=dim\ N$. Let $(U,\phi)$ be a coordinate system on $N$ such that $\phi(U)= \mathbb{R}^d$. Since $\psi$ maps $M$ onto $N$, $\phi\circ\psi(M)=\mathbb{R}^d$. Now I am supposed to get a contradiction from this by proving range $\phi\circ\psi$ has measure zero in $\mathbb{R}^d$, or by any other means (for which the second countability of $M$ is crucial). But I am not able to get this.

Any help will be appreciated!

Solution 1:

Suppose we had $\dim M < \dim N$. Then for every $m \in M$ there are coordinate neighbourhoods $U_m$ of $m$ and $V_m$ of $\psi(m)$ such that in those charts, $\psi$ is the embedding $x \mapsto (x,0)$ of $\mathbb{R}^{\dim M}$ into $\mathbb{R}^{\dim N} \cong \mathbb{R}^{\dim M} \times \mathbb{R}^{\dim N - \dim M}$. Let $K_m$ be the subset of $U_m$ corresponding to the closed unit ball of $\mathbb{R}^{\dim M}$. Then $\psi(K_m)$ is a compact subset of $N$ with empty interior. Since $M$ is second countable, it has a countable basis $\mathcal{B} = \{ B_n : n\in \mathbb{N}\}$ of open sets each of which is contained in some such $K_m$. But then

$$\psi(M) = \psi \left(\bigcup_{n=0}^\infty \overline{B_n}\right) = \bigcup_{n=0}^\infty \psi(\overline{B_n})$$

is a countable union of compact sets with empty interior. Since $N$ is locally compact, hence a Baire space, it follows that $\psi(M)$ has empty interior, and in particular, $\psi$ is not surjective.

Thus if we have a surjective immersion $\psi\colon M \to N$, we must have $\dim M = \dim N$, and such an immersion is a local diffeomorphism. If it is bijective, it is hence a diffeomorphism.