Version of Invariance of Domain for n-manifolds

Question

I am working on the following exercise from Lawson's Topology: A Geometric Approach:

Apply Invariance of Domain

(If $U$ is an open subset of $\mathbb{R}^n$ and $f:U\rightarrow\mathbb{R}^n$ is $1$-$1$ and continuous, then $f$ is an open map)

to prove that if $M$ and $N$ are $n$-manifolds and $f:M\rightarrow N$ is $1$-$1$ and continuous then $f$ is an open map.

I am really stuck. I can't see how to get a map from an open set in $\mathbb{R}^n$ to $\mathbb{R}^n$ involved to apply Invariance of Domain. Any help is greatly appreciated.

Answer 1

Hint: You want to show that $f(U)$ is open in $N$ for all open $U$ in $M$. Note that as $f$ is continuous, for all $x\in U$, there are coordinate charts $U_x \subset U$ so that $f(U_x)$ lie in a coordinate chart $V_x$ of $N$. And you have

$$U = \bigcup_{x\in U} U_x$$