Halo everyone. I would like to enquire how do I solve this question which I extract from Cohn book on Measure Theory.
Let $X$ be a compact Hausdorff space, and let $C(X)$be the set of all real-valued continuous funtions on $X$. Then $B_{o}(X)$, the Baire $\sigma$-algebra on $X$ is the smallest $\sigma$-algebra on $X$ that makes each function in $C(X)$ measurable; the sets that belong to $B_{o}(X)$ are called the Baire subsets of $X$. A Baire measure on $X$ is a finite measure on $(X,B_{o}(X))$
(i) Show that $B_{o}(X)$ is the $\sigma$-algebra generated by the closed $G_{\delta}$'s in $X$.
(ii) Show that if the compact Hausdorff space $X$ is second countable, then $B_{o}(X)=B(X)$
Note: $B(X)$ is the Borel $\sigma$-algebra.
Other question not from the text:
(i) What is the distinction between a locally compact Hausdorff space and a compact Hausdorff space?