A characterization of $X$ for which spaces $C_p(X)$ are distinguished and its applications

TL;DR

This paper characterizes when the space of continuous functions with pointwise convergence topology is distinguished, linking it to the concept of elta-spaces, and explores various applications and properties related to this characterization.

Contribution

It provides a complete characterization of distinguished $C_p(X)$ spaces via elta-spaces and applies this to various classes of topological spaces, including compact and ordinal spaces.

Findings

$C_p(X)$ is distinguished iff $X$ is a elta-space.

If $X$ is ch-complete and $C_p(X)$ is distinguished, then $X$ is scattered.

For certain compact spaces, $C_p(X)$ is distinguished; for others, it is not.

Abstract

We prove that the locally convex space $C_{p}(X)$ of continuous real-valued functions on a Tychonoff space $X$ equipped with the topology of pointwise convergence is distinguished if and only if $X$ is a $\Delta$-space in the sense of \cite {Knight}. As an application of this characterization theorem we obtain the following results: 1) If $X$ is a \v{C}ech-complete (in particular, compact) space such that $C_p(X)$ is distinguished, then $X$ is scattered. 2) For every separable compact space of the Isbell--Mr\'owka type $X$, the space $C_p(X)$ is distinguished. 3) If $X$ is the compact space of ordinals $[0,\omega_1]$, then $C_p(X)$ is not distinguished. We observe that the existence of an uncountable separable metrizable space $X$ such that $C_p(X)$ is distinguished, is independent of ZFC. We explore also the question to which extent the class of $\Delta$-spaces is invariant under basic topological operations.