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Measurements and $G_\delta$-Subsets of Domains

  Published:2010-12-31
 Printed: Jun 2011
  • Harold Bennett,
    Mathematics Department, Texas Tech University, Lubbock, TX, 79409
  • David Lutzer,
    Mathematics Department, College of William and Mary, Williamsburg, VA, 23187
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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 $G_\delta$-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, Čech-complete space, Moore space, $\omega_1$, weakly developable space, sharp base, AF-complete domain-representable, Scott-domain-representable, measurement, Burke's space, developable spaces, weakly developable spaces, $G_\delta$-diagonal, Čech-complete space, Moore space, $\omega_1$, weakly developable space, sharp base, AF-complete
MSC Classifications: 54D35, 54E30, 54E52, 54E99, 06B35, 06F99 show english descriptions Extensions of spaces (compactifications, supercompactifications, completions, etc.)
Moore spaces
Baire category, Baire spaces
None of the above, but in this section
Continuous lattices and posets, applications [See also 06B30, 06D10, 06F30, 18B35, 22A26, 68Q55]
None of the above, but in this section
54D35 - Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54E30 - Moore spaces
54E52 - Baire category, Baire spaces
54E99 - None of the above, but in this section
06B35 - Continuous lattices and posets, applications [See also 06B30, 06D10, 06F30, 18B35, 22A26, 68Q55]
06F99 - None of the above, but in this section
 

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