I am reading Frank Warner's "Foundations of Differentiable Manifolds and Lie Groups" I am trying to prove the existence of unique maximal connected integral manifold of a $k$-distribution $\mathfrak{V}$ passing through a point $p \in M$($Theorem 1.64$).
So I am confused in the following
01-What do they mean by slices?
2-differentiable structure they had defined.
3-To prove the second-countability of $K$ why is it sufficent to prove that only countably many slices of $U_i$ are in $K$?
$1-$For any point $p \in M$, we have by the Local Frobenius theorem($Theorem 1.60$) an open neighborhood $U$ of $p$ and local coordinates ${x_1}^i,{x_2}^i,{x_3}^i \dots {x_d}^i$. By slices DO THEY MEAN the integral manifold of the form ${x_{c+j}}^i={c_j}^i$ for some constant c_j \in \mathbb{R}$ $2,3$- I could not understand what is the differentiable strucuture here. It would be great if someone can decode their notations.