Measurements and \(G_\delta\)-subsets of domains (Q2882464)

The authors investigate domains, Scott domains, and the existence of measurements in the sense of Keye Martin.NEWLINENEWLINEThey make use of a space due to 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 \(ker(\mu)=\max(P).\)NEWLINENEWLINEAmong other things they observe 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.NEWLINENEWLINEFurthermore they verify 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.NEWLINENEWLINEThey also prove 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.NEWLINENEWLINEFinally they establish that the kernel of a measurement on a Scott domain can consistently be a normal, separable, non-metrizable Moore space.