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), 307324 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. ArhangelâskiÇ 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 $\mathfrak {X}= \{ 0\} \cup \{ {n^{  1}}:n= 1,2, \ldots \}$.References

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Additional Information
 © Copyright 1991 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 328 (1991), 307324
 MSC: Primary 54C35; Secondary 57N17, 57N20
 DOI: https://doi.org/10.1090/S0002994719911065602X
 MathSciNet review: 1065602