I recently learned the notion of coordinate chart: If $M$ is a manifold and $U\subseteq M$ is an open set in $M$ then a coordinate chart would be a smooth homeomorphism $\varphi : U \to V \subseteq \mathbb R^n$.
So far so good. Now let $M$ be the smooth manifold $\mathbb R^3$. We can specify a point on $M$ by coordinates of the form
$$ x = r \sin \theta \cos \varphi$$
$$ y = r \sin \theta \sin \varphi$$
$$ z = r \cos \theta $$
where $r \in [0,\infty), \theta \in [0,\pi], \varphi \in [0, 2\pi )$. So the map
$$ \phi (r,\theta, \varphi) = (x,y,z)$$
would be a (global) chart. Except: its domain is not an open subset of $\mathbb R^3$ so it is not a chart. It is however a local chart: for $r \in (0,\infty), \theta \in (0,\pi), \varphi \in (0, 2\pi )$.
Yet, I have seen that $\phi$ is commonly referred to as coordinates which I understand to be a synonym for "coordinate chart".
My question: Is $\phi$ (the spherical coordinates) a coordinate chart or not?