This is the proof of Theorem 5.8 the local slice criterion for embedded submanifolds from John Lee's Introduction to Smooth Manifolds. In this proof, why does the author consider coordinate balls $U_0\subset U$ and $V_0 \subset V$? From the inclusion map and its coordinate representation, isn't it already true that $U=\iota(U)$ is exactly a single slice in $V$? What goes wrong if we don't restrict it to coordinate balls?
Best Answer
The point @TedShifrin is making is that the rank theorem only guarantees the relation $U\subset V$, hence it may not be the case that $V\cap S=U$. In particular, you require that every point in $V\cap S$ has coordinate representation
$$ (x^1,\ldots,x^k,0,\ldots,0), $$
but only the points in $U\subset V\cap S$ have this coordinate representation (the points in $(V\cap S)\setminus U$ could look different).
Edit: Also, Lee defines $S$ to satisfy the local $k$-slice condition if every point in $S$ has a chart $V$ in $M$ such that $V\cap S$ is a single $k$-slice.
Essentially, by restricting to coordinate balls $U_0$ and $V_0$ contained inside of $U$ and $V$, respectively, we guarantee that $U_0$ is a single $k$-slice in $V_0$. By defining the set $V_1$, we guarantee that $V_1\cap S=U_0$.