Domain representability of certain function spaces
A topological space is domain representable if it is homeomorphic to the subspace of maximal elements of a domain, topologized with the Scott topology. Continuing their systematic investigation of domain representability, the authors study this property in the space \(C_p(X)\) of continuous real-valued functions with the topology of pointwise convergence. Motivated by a result of Tkachuk on subcompactness of \(C_p(X)\), they conjecture that for a Tychonoff space \(X\), \(C_p(X)\) is domain representable iff \(X\) is discrete. The main theorem of the paper is a partial result confirming the conjecture for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. As domain representability is closely related to various completeness properties associated with topological games, the authors also characterize the effect of these games on \(C_p(X)\) for a completely regular pseudonormal \(X\). One of the completeness properties studied is strong Choquet completeness which is associated with the so-called strong Choquet game. As a corollary of previous results, it is shown that there exists a GO-space \(X\) such that \(C_p(X)\) is strongly Choquet complete and yet it is not domain representable.
scientific article; zbMATH DE number 5573805