Metrizable quotients of $C_p$-spaces

TL;DR

This paper investigates conditions under which the space of continuous functions with the pointwise topology, $C_p(X)$, admits infinite-dimensional metrizable quotients, extending classical results and constructing specific examples under set-theoretic assumptions.

Contribution

It establishes new criteria for the existence of infinite-dimensional metrizable quotients of $C_p(X)$ and constructs examples under $ eg$CH, improving previous results by Kąkol and Śliwa.

Findings

$C_p(X)$ has an infinite-dimensional metrizable quotient if $X$ contains an infinite discrete $C^*$-embedded subspace.

$C_p(X)$ admits such a quotient if $X$ has a sequence of compact sets with specific embedding properties.

Under $ eg$CH, there exists a zero-dimensional Efimov space with a $C_p$-space having an infinite-dimensional metrizable quotient.

Abstract

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.