Representability of locally compact regular spaces by domains and formal spaces

Abstract

When computing on a (generally) uncountable topological structure gne; , such as the topological field of real numbers R, one must by necessity compute on a set of concrete approximations P for gne; . One way to do this is to represent the original structure gne; using P in such a way that computations on P transfer to approximate computations on gne; . Two such representations are considered, domain representability and representability by formal spaces, and these are compared for the class of locally compact regular spaces. It is shown that for locally compact regular spaces the two representations are equivalent over P for natural sets of approximations P. In addition it is shown that under rather general conditions, a continuous function between topological spaces represented by formal spaces over P 1 and P 2, respectively, lifts to a continuous function between the corresponding domains, the ideal completions of p 1 and p 2.