Problem 2. A comprehensive course in Analysis. Barry simon. Page 239. 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) $\bigcup_{P\in\mathcal{P}}P=X$
Given any continuos function $f$, on a compact Hausdorff space and any $\epsilon>0$, find  a Baire partition $\left\{P_j\right\}_{j=1}^{n}$ so that $\sup_{x,y\in P_j} |f(x)-f(y)|<\epsilon$.
Hint: First find an open cover  by Baire sets, $\left\{U_l\right\}_{l=1}^{n}$ so that for each $l,\ \sup_{x,y\in P_j}|f(x)-f(y)|<\epsilon.$
I do not know how to do this problem. Some help.?
 A: Using the hint, our plan is to first find a finite open cover, followed by a finite compact $G_\delta$ cover and finally a Baire partition.
Let $\varepsilon > 0$ and $x \in X$. By continuity of $f$, we can find an open $U_x$ containing $x$ such that for all $u \in U_x$ we have
$$|f(x) - f(u)| < \frac{\varepsilon}{4}$$
This implies for all $y, z \in U_x$ we get
$$\begin{align}
|f(y) - f(z)| &\le |f(y) - f(x)| + |f(x) - f(z)| \\
&< \frac{\varepsilon}{4} + \frac{\varepsilon}{4} \\
&= \frac{\varepsilon}{2}
\end{align}$$
By order preserving property of supremum, we obtain
$$\sup_{y, z \in U_x} |f(y) - f(z)| \le \frac{\varepsilon}{2} < \varepsilon$$
Next, since $X$ is locally compact Hausdorff, we can choose a compact neighbourhood $K_x$ and open $V_x$ such that $$x \in V_x \subseteq K_x \subseteq U_x$$
Now by lemma 3 of this question (please check!),
we can pick a compact $G_\delta$ set $G_x$ such that $$x \in V_x \subseteq K_x \subseteq G_x \subseteq U_x$$
Observe that $\{V_x\}_{x \in X}$ is an open cover for $X$.
By compactness of $X$, there is a finite subcover $\{V_{x_1}, \dots, V_{x_n}\}$. Since each $V_{x_j} \subseteq G_{x_j}$, $\{G_{x_1}, \dots, G_{x_n}\}$ is also a finite cover. Hence $\{G_{x_1}, \dots, G_{x_n}\}$ is a finite compact $G_\delta$ cover. Besides, since each $G_{x_j} \subseteq U_{x_j}$, we have $$\sup_{y, z \in G_{x_j}} |f(y) - f(z)| < \varepsilon$$
Finally, let $$\begin{align}
\mathcal{P} = \{&B_1 \cap \dots \cap B_n \mid \\
&B_j = G_{x_j} \text{ or } X \setminus G_{x_j} \text{ with } B_1 \cap \dots \cap B_n \neq \emptyset\}
\end{align}$$
We can see that $\mathcal{P}$ is a Baire partition. Fix $B_1 \cap \dots \cap B_n \in \mathcal{P}$. We have $B_1 \cap \dots \cap B_n \subseteq G_{x_j}$ for some $G_{x_j}$. If not, we would have $$B_1 \cap \dots \cap B_n = (X \setminus G_{x_1}) \cap \dots \cap (X \setminus G_{x_n}) = \emptyset$$ contradicting $B_1 \cap \dots \cap B_n$ being nonempty. Hence $$\begin{align}
\sup_{y, z \in B_1 \cap \dots \cap B_n} |f(y) - f(z)| &\le \sup_{y, z \in G_{x_j}} |f(y) - f(z)| \\
&< \varepsilon
\end{align}$$
Done!
