Suppose we know that $f : M \rightarrow N$ is a homeomorphism.
And we are given $g : M \rightarrow N$ that we know is injective and smooth.
Does it follow that $g$ is also bijective? And how would we prove this?
I am also most interested in the case where $M$ and $N$ are smooth manifolds of the same dimension.
The function $$\begin{array}{l|rcl} f : & \mathbb R & \longrightarrow & \mathbb R_+\\ & x & \longmapsto & e^x \end{array}$$
is an homeomorphism.
$$\begin{array}{l|rcl} g : & \mathbb R & \longrightarrow & \mathbb R_+\\ & x & \longmapsto & e^x +1\end{array}$$
is smooth an injective, but not bijective.