Function spaces of low Borel complexity

Abstract

In this paper we investigate situations in which the space C π ( X ) {C_\pi }(X) of continuous, real-valued functions on X X is a Borel subset of the product space R X {{\mathbf {R}}^X} . We show that for completely regular, nondiscrete spaces, C π ( X ) {C_\pi }(X) cannot be a G δ {G_\delta } , an F σ {F_\sigma } , or a G δ σ {G_{\delta \sigma }} subset of R X {{\mathbf {R}}^X} , but it can be an F σ δ {F_{\sigma \delta }} or G δ σ δ {G_{\delta \sigma \delta }} subset.