Measurements and Gδ-subsets of domains

Harold Bennett, David Lutzer

Research output: Contribution to journalArticle

Abstract

In this paper we study domains, Scott domains, and the existence of measurements. We use a space created by D. K. Burke to show that there is a Scott domain P for which max(P) is a G δ-subset of P and yet no measurement μ on P has ker(μ) = max(P). We also correct a mistake in the literature asserting that [0, ω 1) is a space of this type. We show that if P is a Scott domain and X ⊆ max(P) is a G δ- subset of P, then X has a G δ-diagonal and is weakly developable. We show that if X ⊆ max(P) is a G δ-subset of P, where P is a domain but perhaps not a Scott domain, then X is domain-representable, first-countable, and is the union of dense, completely metrizable subspaces. We also show that there is a domain P such that max(P) is the usual space of countable ordinals and is a G δ-subset of P in the Scott topology. Finally we show that the kernel of a measurement on a Scott domain can consistently be a normal, separable, non-metrizable Moore space.

Original languageEnglish
Pages (from-to)193-206
Number of pages14
JournalCanadian Mathematical Bulletin
Volume54
Issue number2
DOIs
StatePublished - Jun 2011

Keywords

  • AF-complete
  • Burke's space
  • Developable spaces
  • Domain-representable
  • G -diagonal
  • Measurement
  • Moore space
  • Scott-domain-representable
  • Sharp base
  • Weakly developable space
  • Weakly developable spaces
  • Čech-complete space
  • ω

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