A topological space $X$ is said to be normal if each point of $X$ is a closed subset of $X$ and for each pair of closed sets $C_{1}, C_{2} \subset X$ with $C_{1} \cap C_{2}=\emptyset$ there are open sets $U_{1}, U_{2} \subset X$ so that $C_{i} \subset U_{i}$ and $U_{1} \cap U_{2}=\emptyset$. In this case we say that the $U_{i}$ separate the $C_{i}$.

Show that a compact Hausdorff space is normal. [Hint: first consider the case where $C_{2}$ is a point.]

For $C \subset X$ we define an equivalence relation $\sim_{C}$ on $X$ by $x \sim_{C} y$ for all $x, y \in C$, $x \sim_{C} x$ for $x \notin C$. If $C, C_{1}$ and $C_{2}$ are pairwise disjoint closed subsets of a normal space $X$, show that $C_{1}$ and $C_{2}$ may be separated by open subsets $U_{1}$ and $U_{2}$ such that $U_{i} \cap C=\emptyset$. Deduce that the quotient space $X / \sim_{C}$ is also normal.