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

英文摘要：

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 $Δ$-space in the sense of \cite {Knight}. As an application of this characterization theorem we obtain the following results: 1) If $X$ is a Čech-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--Mrówka type $X$, the space $C_p(X)$ is distinguished. 3) If $X$ is the compact space of ordinals $[0,ω_1]$, 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 $Δ$-spaces is invariant under basic topological operations.