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

Dobrowolski, T.; Marciszewski, W.; Mogilski, J.

doi:10.1090/S0002-9947-1991-1065602-X

On topological classification of function spaces $C_ p(X)$ of low Borel complexity
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by T. Dobrowolski, W. Marciszewski and J. Mogilski PDF

Trans. Amer. Math. Soc. 328 (1991), 307-324 Request permission

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 $\mathfrak {X}= \{ 0\} \cup \{ {n^{ - 1}}:n= 1,2, \ldots \}$.

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