Problem 1.11

Let $M={\overline{𝔹}}^n$, the closed unit ball in ${ℝ}^n$. Show that $M$ is a topological manifold with boundary in which each point in ${𝕊}^{n-1}$ is a boundary point and each point in ${𝔹}^n$ is an interior point. Show how to give it a smooth structure such that every smooth interior chart is a smooth chart for the standard smooth structure on ${𝔹}^n$. [Hint: consider the map $\pi \circ {\sigma }^{-1}:{ℝ}^n\to {ℝ}^n$, where $\sigma :{𝕊}^n\to {ℝ}^n$ is the stereographic projection (Problem 1-7) and $\pi$ is a projection from ${ℝ}^{n+1}$ to ${ℝ}^n$ that omits some coordinate other than the last.]