© 1996, Topology Atlas

Intuition from Domain Theory, where the Scott topology plays a central role, suggests to consider quasi-uniform spaces together with an additional topology which may, but need not be the induced one. This idea goes back to Mike Smyth who introduced in [1] suitable axioms for topologies accompanying quasi-uniform spaces. This setting of topological quasi-uniform spaces allows to develop a theory of completeness and completion ([2], [3]). The class of topological quasi-uniform spaces contains all posets (with the Alexandroff topology and the trivial quasi-uniformity); for such a space Smyth-completion coincides with ideal completion, the topology on the completed space being its Scott-topology. On the other hand, Smyth-completion of a uniform spaces (in its induced topology) simplifies to its usual Cauchy completion. Applications of this theory concentrate on the origin of the subject: The ongoing foundational work on quantitative domain theory, where the spaces are utilized to serve as domains for denotational semantics of programming languages.

[1] M.B. Smyth. Quasi-uniformities: Reconciling domains with metric spaces. In Third Workshop on Mathematical Foundations of Programming Language Semantics, volume 298 of Lecture Notes in Computer Science, pages 236--253. Springer Verlag, 1988.

[2] M.B. Smyth. Completeness of quasi-uniform and syntopological spaces. Journal of the London Mathematical Society, 49(2):385--400, 1994.

[3] Ph. Sünderhauf. The Smyth-completion of a quasi-uniform space. In M. Droste and Y. Gurevich, editors, Semantics of Programming Languages and Model Theory, volume 5 of "Algebra, Logic and Applications'', pages 189--212. Gordon and Breach Science Publ., 1993.