Exercise 4, § 3

A completely regular space $X$ is such that the metrizable space $\mathscr{C}_u(X; \mathbf{R})$ is of countable type if (and only if) $X$ is compact and metrizable. [By considering the Stone-Čech compactification of $X$ (Chapter IX, § 1, Exercise 7) show that $X$ must be metrizable, and then observe that the space $\mathscr{C}_u(Z; \mathbf{R})$ is not of countable type.]