  The famous Rosenthal-Lacey theorem asserts that for each infinite compact set
$K$ the Banach space $C(K)$ admits a quotient which is either a copy of $c$ or
$\ell_{2}$. What is the case when the uniform topology of $C(K)$ is replaced by
the pointwise topology? Is it true that $C_p(X)$ always has an
infinite-dimensional separable (or better metrizable) quotient? In this paper
we prove that for a Tychonoff space $X$ the function space $C_p(X)$ has an
infinite-dimensional metrizable quotient if $X$ either contains an infinite
discrete $C^*$-embedded subspace or else $X$ has a sequence $(K_n)_{n\in\mathbb
N}$ of compact subsets such that for every $n$ the space $K_n$ contains two
disjoint topological copies of $K_{n+1}$. Applying the latter result, we show
that under $\lozenge$ there exists a zero-dimensional Efimov space $K$ whose
function space $C_{p}(K)$ has an infinite-dimensional metrizable quotient.
These two theorems essentially improve earlier results of K\k{a}kol and \'Sliwa
on infinite-dimensional separable quotients of $C_p$-spaces.

General Topology;Functional Analysis