Measurements and G δ-Subsets of Domains

Bennett, Harold ; Lutzer, David

Canadian mathematical bulletin, Tome 54 (2011) no. 2, pp. 193-206

Résumé

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 ${{G}_{\delta }}$ -subset of $P$ and yet no measurement $\mu $ on $P$ has $\text{ker(}\mu \text{)}\,=\,\max (P)$ . We also correct a mistake in the literature asserting that $[0,\,{{\omega }_{1}})$ is a space of this type. We show that if $P$ is a Scott domain and $X\,\subseteq \,\max (P)$ is a ${{G}_{\delta }}$ -subset of $P$ , then $X$ has a ${{G}_{\delta }}$ -diagonal and is weakly developable. We show that if $X\,\subseteq \,\max (P)$ is a ${{G}_{\delta }}$ -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 ${{G}_{\delta }}$ -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.

Détail

DOI : 10.4153/CMB-2010-104-3

Mots-clés : 54D35, 54E30, 54E52, 54E99, 06B35, 06F99, domain-representable, Scott-domain-representable, measurement, Burke's space, developable spaces, weakly developable spaces, G δ-diagonal, Čech-complete space, Moore space, ω1, weakly developable space, sharp base, AF-complete