Measurements and G(delta)-Subsets of Domains
Canadian Mathematical Bulletin-Bulletin Canadien DE Mathematiques
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 subset of 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 subset of 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 Pin 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.
Bennett, Harold and Lutzer, David, Measurements and G(delta)-Subsets of Domains (2011). Canadian Mathematical Bulletin-Bulletin Canadien DE Mathematiques, 54(2), 193-206.