A metric space of A. H. Stone and an example concerning 𝜎-minimal bases

Abstract

In this paper we use a metric space Y Y due to A. H. Stone and one of its completions X X to construct a linearly ordered topological space E = E ( Y , X ) E = E(Y,X) that is Čech complete, has a σ \sigma -closed-discrete dense subset, is perfect, hereditarily paracompact, first-countable, and has the property that each of its subspaces has a σ \sigma -minimal base for its relative topology. However, E E is not metrizable and is not quasi-developable. The construction of E ( Y , X ) E(Y,X) is a point-splitting process that is familiar in ordered spaces, and an orderability theorem of Herrlich is the link between Stone’s metric space and our construction.