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 G δ -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 G δ -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 G δ -subsets. In addition, it follows that if ( X , τ ) is a subcompact GO-space constructed on any set of real numbers and if τ S is the topology obtained from τ by isolating all points of a set S ⊆ X , then ( X , τ S ) is also subcompact. Whether these two assertions hold for arbitrary subcompact spaces is not known. 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 G δ -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 G δ -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.