The Gruenhage property, property*, fragmentability, and σ-isolated networks in generalized ordered spaces

Harold Bennett, David Lutzer

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We examine the Gruenhage property, property (introduced by Orihuela, Smith, and Troyanski), fragmentability, and the existence of σ-isolated networks in the context of linearly ordered topological spaces (LOTS), generalized ordered spaces (GO- spaces), and monotonically normal spaces. We show that any monotonically normal space with property or with a σ-isolated network must be hereditarily paracompact, so that property and the Gruenhage property are equivalent in monotonically normal spaces. (However, a fragmentable monotonically normal space may fail to be paracompact.) We show that any fragmentable GO-space must have a σ-disjoint π-base and it follows from a theorem of H. E. White that any fragmentable, first-countable GO-space has a dense metrizable subspace. We also show that any GO-space that is fragmentable and is a Baire space has a dense metrizable subspace.We show that in any compact LOTS X, metrizabil- ity is equivalent to each of the following: X is Eberlein compact; X is Talagrand compact; X is Gulko compact; X has a σ-isolated network; X is a Gruenhage space; X has prop- erty : X is perfect and fragmentable; the function space C(X) has a strictly convex dual norm. We give an example of a GO-space that has property , is fragmentable, and has a σ-isolated network and a σ-disjoint π-base but contains no dense metrizable subspace.

Original languageEnglish
Pages (from-to)273-294
Number of pages22
JournalFundamenta Mathematicae
Volume223
Issue number3
DOIs
StatePublished - 2013

Keywords

  • Dense metrizable subspace
  • Eberlein compact
  • Fragmentable space
  • G-diagonal
  • GO-space
  • Generalized ordered space
  • Gruenhage space
  • Gulko compact
  • LOTS
  • Linearly ordered topological space
  • Metriz-Ability
  • Michael line
  • Monotone normality
  • Paracompactness
  • Property
  • Quasi-developable space
  • Sorgenfrey line
  • Stationary sets
  • Strictly convex dual norm
  • Talagrand compact
  • σ-disjoint π-base
  • σ-isolated net-work

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