Measurements and $G_delta$-Subsets of Domains

Canadian mathematical bulletin

Harold Bennett¹  David Lutzer² 

[1] Mathematics Department, Texas Tech University, Lubbock, TX, 79409;Mathematics Department, College of William and Mary, Williamsburg, VA, 23187

关键词: domain-representable;    Scott-domain-representable;    measurement;    Burke's space;    developable spaces;    weakly developable spaces;    $G_delta$-diagonal;    ?ech-complete space;    Moore space;    $omega_1$;    weakly developable space;    sharp base;    AF-complete;   

DOI  :  10.4153/CMB-2010-104-3

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

In this paper we study domains, Scottdomains, and the existence of measurements. Weuse a space created by D.~K. Burke to show thatthere is a Scott domain $P$ for which $max(P)$ isa $G_delta$-subset of $P$ and yet no measurement$mu$ on $P$ has $ker(mu) = max(P)$. We alsocorrect 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.

【 授权许可】

Unknown