Perfectly normal non-metrizable non-Archimedean spaces are generalized Souslin lines

Abstract

In this paper we prove the equivalence between the existence of perfectly normal, non-metrizable, non-archimedean spaces and the existence of “generalized Souslin lines", i.e., linearly ordered spaces in which every collection of disjoint open intervals is σ \sigma -discrete, but which do not have a σ \sigma -discrete dense set. The key ingredient is the observation that every first countable linearly ordered space has a dense non-archimedean subspace.