## 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 language | English |
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Pages (from-to) | 193-206 |

Number of pages | 14 |

Journal | Canadian Mathematical Bulletin |

Volume | 54 |

Issue number | 2 |

DOIs | |

State | Published - 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