Consistent Smyth powerdomains of topological spaces and quasicontinuous domains

Abstract

The consistent Smyth powerdomain RQC(X) of a topological space X means the family of all nonempty relatively compact-connected saturated subsets of X, ordered by the reverse inclusion and endowed with the upper Vietoris topology. In this paper, we study properties of consistent Smyth powerdomains of certain topological spaces (especially, quasicontinuous domains equipped with the Scott topology) from topological, order theoretical and categorical aspects. Main results are: (i) a topological space X is sober iff RQC(X) is sober; (ii) if X is locally compact-connected, well-filtered and coherent, then RQC(X) is coherent; (iii) every dcpo equipped with the Scott topology is locally connected, and if a dcpo L is finitely up-generated, locally compact, well-filtered and coherent, then RQC(L) is a Lawson compact L-domain; (iv) if L is a quasicontinuous domain (resp., a Lawson compact quasicontinuous domain, a quasialgebraic domain), then RQC(L) is a continuous dcpo-∧↑-semilattice (resp., a Lawson compact L-domain, an algebraic dcpo-∧↑-semilattice); (v) it is proved that RQC(L) is a free continuous dcpo-∧↑-semilattice over a quasicontinuous domain L.