Exercise 9, § 3

Let K be a metrizable compact space, $f$ a continuous mapping of K into a Hausdorff topological space E, and $k$ any integer $> 1$. Denote by $B_k$ the subset of E formed by the $y$ such that $f^{-1}(y)$ contains at least $k$ distinct points; show that $A_k = f^{-1}(B_k)$ is a Borel set in K (for every integer $n > 0$, let $B_{kn}$ be the subset of E formed by the $y$ such that $f^{-1}(y)$ contains at least $k$ points whose mutual distances are all $\geq 1/n$; show that $B_{kn}$ is closed).