Maximal Limit Spaces, Powerspaces, and Scott Domains

Abstract

We explore an area that connects classical Hausdorff topology and the Scott domain theory and serves as a foundation for a denotational semantics of numerical programs.Our key notion is that of a maximal limit space, a T0 space (X, T ) in which every net that has a limit point has a unique limit point maximal in the specialization order induced by T . Maximal limit spaces combine features of Hausdorff spaces and domains and form a bridge between those two categories. Every Hausdorff space is a maximal limit space, and maximal limit spaces are preserved under product, closed subspace, and function space constructions. A topological version of the lifting construction, familiar in domain theory, makes a maximal limit space into a compact maximal limit space. The upper powerspace construction makes a locally compact maximal limit space into a c.b.c. domain (continuous directed-complete partial order that is bounded-complete, i.e., any subset with an upper bound has a least upper bound) that is pointed (has a bottom element) if the original space was compact. C.b.c. domains are locally compact maximal limit spaces. The space of continuous functions from a locally compact topological space to a pointed c.b.c. domain is a pointed c.b.c. domain. The topology of pointwise convergence and the compact-open topology are identical on such function spaces.The upper powerspace construction is functorial and behaves well in relation to function space formation.