I am having difficulty understanding the proof below from John Lee's Introduction to Smooth Manifolds.
First, what exactly is the projection map $\pi_v$? I can't see how a projection map can be defined by having the kernel $\mathbb{R}v$.
Finally, the correction on the book states that this argument does not apply when $M$ has nonempty boundary because in that case $M\times M$ is not a smooth manifold with boundary. But we can consider the restrictions to $\kappa$ to $(M\times Int M)\setminus \Delta_m$ and to $(M \times \partial M)\setminus \Delta_M$ (both of which are smooth manifolds with boundary), and note that there is a point $[v]\in \mathbb{RP}^{N-1}$ that is not in the image of $\tau $ or either of these restrictions of $\kappa$.
How do we know that there is such a point $[v]\in \mathbb{RP}^{N-1}$? And why are $(M\times Int M)\setminus \Delta_M$ and $(M \times \partial M)\setminus \Delta_M$ and $TM \setminus M_0$ smooth manifolds with boundary? I know each of $M\times Int M$, $M \times \partial M$ and $TM$ are but how do we ensure they are still smooth manifolds with boundary minus $\Delta_M$ and $M_0$? Is it simply because they become open subsets?
