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Search: MSC category 06B35 ( Continuous lattices and posets, applications [See also 06B30, 06D10, 06F30, 18B35, 22A26, 68Q55] )

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1. CMB 2010 (vol 54 pp. 193)

Bennett, Harold; Lutzer, David
 Measurements and $G_\delta$-Subsets of Domains 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 $\ker(\mu) = \max(P)$. 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. Keywords:domain-representable, Scott-domain-representable, measurement, Burke's space, developable spaces, weakly developable spaces, $G_\delta$-diagonal, Äech-complete space, Moore space, $\omega_1$, weakly developable space, sharp base, AF-completeCategories:54D35, 54E30, 54E52, 54E99, 06B35, 06F99
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