Subcompactness and domain representability in GO-spaces on sets of real numbers

In this paper we explore a family of strong completeness properties in GO-spaces defined on sets of real numbers with the usual linear ordering. We show that if τ is any GO-topology on the real line R, then (R,τ) is subcompact, and so is any Gδ-subspace of (R,τ). We also show that if (X,τ) is a subcompact GO-space constructed on a subset X⊆R, then X is a Gδ-subset of any space (R,σ) where σ is any GO-topology on R with τ=σ|X. It follows that, for GO-spaces constructed on sets of real numbers, subcompactness is hereditary to Gδ-subsets. In addition, it follows that if (X,τ) is a subcompact GO-space constructed on any set of real numbers and if τS is the topology obtained from τ by isolating all points of a set S⊆X, then (X,τS) is also subcompact. Whether these two assertions hold for arbitrary subcompact spaces is not known.We use our results on subcompactness to begin the study of other strong completeness properties in GO-spaces constructed on subsets of R. For example, examples show that there are subcompact GO-spaces constructed on subsets X⊆R where X is not a Gδ-subset of the usual real line. However, if (X,τ) is a dense-in-itself GO-space constructed on some X⊆R and if (X,τ) is subcompact (or more generally domain-representable), then (X,τ) contains a dense subspace Y that is a Gδ-subspace of the usual real line. It follows that (Y,τ|Y) is a dense subcompact subspace of (X,τ). Furthermore, for a dense-in-itself GO-space constructed on a set of real numbers, the existence of such a dense subspace Y of X is equivalent to pseudo-completeness of (X,τ) (in the sense of Oxtoby). These results eliminate many pathological sets of real numbers as potential counterexamples to the still-open question: “Is there a domain-representable GO-space constructed on a subset of R that is not subcompact”? Finally, we use our subcompactness results to show that any co-compact GO-space constructed on a subset of R must be subcompact.