Metrizable Bounded Sets in C(X) Spaces and Distinguished C_(p)(X) Spaces

Quite recently W.\,Ruess [{\it Locally convex spaces not containing $\ell_{1}$}, Funct. Approx. Comment. Math. 50 (2014) 351--358] has shown that a wide class of locally convex spaces for which all bounded sets are metrizable enjoy Rosenthal's $\ell_{1}$-dichotomy. Being motivated by this fact we show that for a Tychonoff space $X$ the bounded sets of $C_{p}(X)$ are metrizable (respectively, the bounded sets of $C_{k}(X)$ are weakly metrizable) if and only if $X$ is countable. If $X$ is a $P$-space we show that every bounded set in $C_{p}(X)$ is metrizable if and only if $X$ is countable and discrete. The second part of the paper deals with distinguished $C_{p}(X)$ spaces. Among other things we show that $C_{p}(X)$ is distinguished if and only if the strong topology of the dual coincides with its strongest locally convex topology, and that $C_{p}(X)$ is always distinguished whenever $X$ is countable.