The β-space property in monotonically normal spaces and GO-spaces

In this paper we examine the role of the beta-space property (equivalently of the MCM-property) in generalized ordered (GO-)spaces and, more generally, in monotonically normal spaces. We show that a GO-space is metrizable iff it is a beta-space with a G(delta)-diagonal and iff it is a quasi-developable beta-space. That last assertion is a corollary of a general theorem that any beta-space with a sigma-point-finite base must be developable. We use a theorem of Balogh and Rudin to show that any monotonically normal space that is hereditarily monotonically countably metacompact (equivalently, hereditarily a beta-space) must be hereditarily paracompact, and that any generalized ordered space that is perfect and hereditarily a beta-space must be metrizable. We include an appendix on non-Archimedean spaces in which we prove various results announced without proof by Nyikos. (C) 2005 Elsevier B.V. All rights reserved.