On topological classification of function spaces $C_ p(X)$ of low Borel complexity

Abstract:

We prove that if $X$ is a countable nondiscrete completely regular space such that the function space ${C_p}(X)$ is an absolute ${F_{\sigma \delta }}$-set, then ${C_p}(X)$ is homeomorphic to ${\sigma ^\infty }$, where $\sigma = \{ ({x_i}) \in {{\mathbf {R}}^\infty }:{x_i}= 0$ for all but finitely many $i\}$. As an application we answer in the negative some problems of A. V. Arhangel’skiǐ by giving examples of countable completely regular spaces $X$ and $Y$ such that $X$ fails to be a ${b_R}$-space and a $k$-space (and hence $X$ is not a ${k_\omega }$-space and not a sequential space) and $Y$ fails to be an ${\aleph _0}$-space while the function spaces ${C_p}(X)$ and ${C_p}(Y)$ are homeomorphic to ${C_p}(\mathfrak {X})$ for the compact metric space $\mathfrak {X}= \{ 0\} \cup \{ {n^{ - 1}}:n= 1,2, \ldots \}$.