Abstract view
Measurements and $G_\delta$Subsets of Domains


Published:20101231
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 domainrepresentable,
firstcountable, 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,
nonmetrizable Moore space.
Keywords: 
domainrepresentable, Scottdomainrepresentable, measurement, Burke's space, developable spaces, weakly developable spaces, $G_\delta$diagonal, Čechcomplete space, Moore space, $\omega_1$, weakly developable space, sharp base, AFcomplete
domainrepresentable, Scottdomainrepresentable, measurement, Burke's space, developable spaces, weakly developable spaces, $G_\delta$diagonal, Čechcomplete space, Moore space, $\omega_1$, weakly developable space, sharp base, AFcomplete

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
