On terminology related to the definition of diffeomorphism

Question

I have seen something like if $U$ and $V$ are open subsets of $\mathbb{R}^n$ then we say $U$ and $V$ are diffeomorphic if there exists a diffeomorphism $f$ from $U$ to $V$. I have only seen it for opens sets. Does it make sense to talk about diffeomorphic sets for closed sets as well?

For example in the above example, say $C$ is a closed subset of $U$, then does it make sense to say $f|_C$ defines a diffeomorphism from $C$ to $D = f(C)$ (and say $C$ is diffeomorphic to $D$)?

Answer 1

The correct definition in that case (without assumption on $C$ and $D$) is

Definition. $C$ and $D$ are diffeomorphic if there are open neighbourhoods $C\subset U$ and $D\subset V$ and a diffeomorphism $\phi:U\to V$ such that $\phi(C)=D$.