Let $M$ be an n-dimensional manifold with atlas $\pi_{\alpha}: U_{\alpha} \rightarrow V_{\alpha}$.
Show that the borel $\sigma$-algebra on $M$ is the unique $\sigma$-algebra whos restriction to each $U_\alpha$ is the pullback via $\pi_\alpha$
So, the Borel $\sigma$-algebra on $M$ is the closure of all the open sets in the topology of $M$ under countable unions, intersections and complements.
Also, I thought the pullback via $\pi_\alpha$ would define a $\sigma$-algebra on the domain of $\pi_\alpha$, which in this case I suppose would be $\mathbb{R}^n$. Even the wording of the problem is strange to me, as we are restricting a $\sigma$-algebra on $M$ to $U_\alpha$, but $U_\alpha$ isn't even actually just $M$! It's just homeomorphic to a hood in $M$. What am i not understanding correctly here? Thanks!