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

Harold R. Bennett, David J. Lutzer

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

In this paper we examine the role of the β-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 β-space with a Gδ-diagonal and iff it is a quasi-developable β-space. That last assertion is a corollary of a general theorem that any β-space with a σ-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 β-space) must be hereditarily paracompact, and that any generalized ordered space that is perfect and hereditarily a β-space must be metrizable. We include an appendix on non-Archimedean spaces in which we prove various results announced without proof by Nyikos.

Original languageEnglish
Pages (from-to)2218-2228
Number of pages11
JournalTopology and its Applications
Volume153
Issue number13
DOIs
StatePublished - Jul 1 2006

Keywords

  • Dense metrizable subspace
  • GO-space
  • Generalized ordered space
  • Hereditarily MCM
  • MCM
  • Metrization
  • Monotonically countably metacompact
  • Monotonically normal
  • Non-Archimedean space
  • Paracompact
  • Perfect space
  • Quasi-developable
  • Stationary set
  • β-space
  • σ-closed-discrete dense set

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