Guillemin and Pollack section 1.4, page 21, corresponding figure 1-12.
Select local coordinates around $x$ and $y$ so that $f(x_1, ...x_k) = (x_1, ...., x_l)$ and $y$ corresponds to $(0, ..,0)$. Thus near $x$, $f^{-1}(y)$ is just the set of points $(0, ...,0, x_{l+1},...,x_k)$.
My understanding of the point $x$ is that it has coordinates $(x_1, ...., x_k)$. My confusion is about the "set of points" in text.
Why are $(0,...,0, x_{l+1},...,x_k)$ points and not coordinates?