Exercise 9, § 4

Let $B$ and $F$ be topological spaces and let $E$ be a locally trivial $B$-space with fiber-type $F$.

(a) Suppose that $B$ is paracompact and that $F$ is homeomorphic to the sum space of a family of compact spaces. Prove that then $E$ is paracompact. (Resume the proof of I, p. 84, Prop. 8.)

(b) Suppose that $B$ and $F$ are metrizable. Prove that the same is true of $E$.

(c) Suppose that $B$ and $F$ are countable at infinity. Prove that $E$ is countable at infinity.