Descriptive complexity of function spaces

Abstract

In this paper we show that C π ( X ) {C_\pi }(X) , the set of continuous, real-valued functions on X X topologized by the pointwise convergence topology, can have arbitrarily high Borel or projective complexity in R X {{\mathbf {R}}^X} even when X X is a countable regular space with a unique limit point. In addition we show how to construct countable regular spaces X X for which C π ( X ) {C_\pi }(X) lies nowhere in the projective hierarchy of the complete separable metric space R X {{\mathbf {R}}^X} .