I am reading Milnor's Topology from a Differentiable Viewpoint, which is very nice. I am reading about regular points versus critical points, etc. I ran across some notation that I was not very clear about. On page 11, as part of the proof of Brown's theorem, he references a particular operator, and I was not clear on what the notation was actually referring to. if anyone can explain, it would really help.
So he is talking about an manifold $M \subset R^k$,$N \subset R^m$ and a linear map $L: R^k \rightarrow R^{m-n}$ that is nonsingular on the subspace $\mathfrak{R} \subset TM_x \subset R^k$.
This is where it gets confusing for me:
Now define $$ F: M \rightarrow N \times R^{m-n} $$
by $F(\xi) = (f(\xi), L(\xi))$ The derivative $dF_x$ is clearly given by the formula
$$ dF_x = (df_x(\nu), L(\nu)) $$
I was trying to understand what this mapping $F: M \rightarrow N \times R^{m-n}$ means, and this reference to $F(\xi) = (f(\xi), L(\xi))$. Is $F$ the cartesian product of all points in N with all vectors in $R^{m-n}$? What would be the use of that? Also, this $F(\xi)$ notation, what does it mean that I have $(f(\xi), L(\xi))$. Like what does that tuple represent, if both of those are mappings to different sized spaces?
So I think I am missing what the intent of the notation is.