Domain representability of certain function spaces

Let C p ( X ) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that C p ( X ) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then C p ( X ) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space C p ( X ) , Oxtobyʹs pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement “every countable subset of X is closed”.