I'm reding Definition 1.4 in this lecture note.
Let $X \subset \mathbb{R}^{N}$. Then we say that $X$ is a (smooth) $k$ (-dimensional) manifold if for all $x \in X, \exists V \subset X$ open (in the topology of $X$ ) such that $V$ is diffeomorphic to some open $U \subset \mathbb{R}^{k}$.
A diffeomorphism $\varphi: U \rightarrow V \subset X$ with $U \subset \mathbb{R}^{k}$ is then called a local parameterisation of the neighbourhood $V .$ The inverse diffeomorphism $\varphi^{-1}: V \rightarrow U$ is called a local coordinate system, or chart, on $V$.
Then there are an illustration and a remark.
Note: Wlog we can assume that $U$ in the above definition is a disc, and hence that $U$ is diffeomorphic to $\mathbb{R}^{k}$. So we can say "such that $V$ is diffeomorphic to $\mathbb{R}^{k}$ " and everything would still work, but the above definition is "more" general.
Not every open set is diffeomorphic to a disc, let alone $\mathbb R^k$. I would like to ask if this remark is correct or If I miss something.
Best Answer
That "Note" does not say $U$ is diffeomorphic to a disc or to $\mathbb R^k$. Instead, what that note says is that if we are given one diagram of sets and maps as illustrated, satisfying all the requirements of that definition, then there exists another diagram that also satisfies all the requirements of that definition together with the additional requirement that $U$ is a disc.
For the proof, simply choose an open disc $D \subset \mathbb R^k$ such that $y \in D \subset U$, then restrict $\phi$ to $D$ and restrict $\phi^{-1}$ to $\phi(D)$. In this new diagram, $D$ becomes the "new $U$" (call it $U'$) and $\phi(D)$ becomes the "new $V$" (call it $V'$).