Representations of Complete Uniform Spaces via Uniform Domains

AbstractIn this paper, we show that complete uniform spaces can be represented domain-theoretically. We introduce the notion of a uniform domain, which is an ω-algebraic domain with some uniform structure on the set K(D) of finite elements of D. It is proved that when (X,μ) is a complete uniform space of countable weight, there is a uniform domain D such that X is the retract of the set L(D) of limit elements of D. On the other hand, in every uniform domain D, there exists a minimal subspace M(D) of L(D) on which K(D) induces a uniformity structure. Thus, a uniform domain can be considered as a set with a particular kind of base of a uniformity. Since every infinite increasing sequences in K(D) identifies one element of M(D), through a labelling of edges of K(D), we obtain an admissible representation of a uniform space in a uniform domain. We also show that such a representation is a proper representation