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?

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    $\begingroup$ You have a typo. Presumably you mean $f(x)=(x_1,\dots,x_\ell)$? To answer your question, any time you have local coordinates on $U\subset X$, you identify points of $U$ with their coordinates in $\Bbb R^k$. I mean, don't you think of $(1,2)\in\Bbb R^2$ as a point in $\Bbb R^2$, even though you're specifying a vector of coordinates? $\endgroup$ – Ted Shifrin Jul 21 '18 at 17:09
  • $\begingroup$ typo indeed, thanks for correcting. I had not understood the term "identify with", used in the aforementioned book. Your comment certainly helps, thanks again. $\endgroup$ – G. Debailly Jul 23 '18 at 8:40

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