Definition (Baire set) Let X be a compact Hausdorff space. The Baire sets are the smallest $\sigma$-algebra containing all compacts $G_{\delta}$'s.
Definition (Partition) Given an algebra , $\mathcal{U}$, a partition associated to $\mathcal{U}$, is a finite subset $\mathcal{P}\subset \mathcal{U}$ so that
(i) All sets in $\mathcal{P}$ are nonempty
(ii) $P_1,\ P_2\in\mathcal{P}\Rightarrow P_1\cap P_2=\emptyset$
(iii) $\bigcup_{P\in\mathcal{P}}P=X$
Problem:
Given any open cover $\left\{U_{a}\right\}_{a\in I}$, of a compact Hausdorff space $X$, prove that one can find a partition $\left\{P_j\right\}_{j=1}^{n}$ into Baire sets so that each $P_j$ lies in some single $U_a$. (Hint: First find an open cover by Baire sets, $\left\{V_l\right\}_{l=1}^{m}$, so each $V_l$ is in some $U_a$.
I have this... Let $\left\{U_{a}\right\}_{a\in I}$ open cover of $X$. $X$ is compact, then exists $1,\ldots, m$ such that $\left\{U_{l}\right\}_{l=1}^{m}$ is finite open cover of $X$.
Now, $X=\bigcup_{l=1}^{m} U_l$, and $\bigcup_{l=1}^{m} U_l=\bigcup_{l=1}^{m} U_l\setminus(U_1\cup \ldots, \cup U_{l-1})$ with $U_{0}=\emptyset.$ now this union is disjoint.
Define $P_l=U_l\setminus (U_1\cup \ldots \cup U_{0})$. This works?...