Generalized Ultrametric Spaces: Completion, Topology, and Powerdomains
by
J.J.M.M. Rutten
CWI
Coauthors: F. van Breugel (Vrije Universiteit), M. M. Bonsangue (CWI)

Generalized versions of ultrametric spaces are considered in which the distance need not be symmetric and where different elements may have distance 0. They provide a common generalization of both preordered spaces and ordinary ultrametric spaces. A number of basic definitions and facts will be given. The completion of a generalized ultrametric space is defined along the lines of Smyth' Cauchy completion of quasi metric spaces. Following the approach of Lawvere in which generalized ultrametric spaces are viewed as [0, 1]-enriched categories, an alternative, equivalent definition of completion is then formulated in terms of a metric version of the Yoneda Lemma, leading to simple proofs of some properties. Next a topology for generalized ultrametric spaces is defined, which combines both the Scott topology on preorders and the ordinary \epsilon-ball topology on ultrametric spaces, in a way that preserves the nice properties of both. It is shown that the closure operator corresponding to this generalized Scott topology, as we have called it, arises naturally via, again, the Yoneda Lemma. This lemma is finally applied once more in the construction of a generalized lower powerdomain. A comparison will be made with related work, amongst others by Smyth, Flagg and Koppermann, and Wagner.

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaf-73.