I am reading "Analysis on Manifolds" by James R. Munkres.
Theorem 18.2. Let $g:A\to B$ be a diffeomorphism of class $C^r,$ where $A$ and $B$ are open sets in $\mathbb{R}^n.$ Let $D$ be a compact subset of $A,$ and let $E=g(D).$
(a) We have $g(\operatorname{Int}D)=\operatorname{Int}E$ and $g(\operatorname{Bd}D)=\operatorname{Bd}E.$
(b) If $D$ is rectifiable, so is $E.$
These results also hold when $D$ is not compact, provided $\operatorname{Bd}D\subset A$ and $\operatorname{Bd}E\subset B.$
Proof. (a) The map $g^{-1}$ is continuous. Therefore, for any open set $U$ contained in $A,$ the set $g(U)$ is an open set contained in $B.$
Of course, the author's proof of this part is right.
But the author wrote the following theorem on p.65.
Theorem 8.2. Let $A$ be open in $\mathbb{R}^n;$ let $f:A\to\mathbb{R}^n$ be of class $C^r;$ let $B=f(A).$ If $f$ is one-to-one on $A$ and if $Df(x)$ is non-singular for $x\in A,$ then the set $B$ is open in $\mathbb{R}^n$ and the inverse function $g:B\to A$ is of class $C^r.$
Let $U$ be an open set contained in $A.$
Then $g:U\to\mathbb{R}^n$ is of class $C^r.$
$g$ is one-to-one on $U.$
$Dg(x)$ is non-singular for $x\in U$.
So, by Theorem 8.2, $g(U)$ is an open set contained in $B$.
The author didn't write as follows.
Why?
Proof. (a) By Theorem 8.2, for any open set $U$ contained in $A,$ the set $g(U)$ is an open set contained in $B.$