Weightable Directed Spaces
by
M. Schellekens
Universität Siegen

The theory of complexity spaces has been introduced in as part of the development of a topological foundation for Complexity Analysis.

The topological study of these spaces has been continued in the context of the theory of the upper weightable spaces (), while the specific properties of total boundedness and Smyth completeness have been analyzed in .

Here we introduce a technique of ``lifting'', which allows one to extend an upper weightable space, and hence a complexity space, by a maximum. This leads to a characterization of the upper weightable spaces as the weightable spaces which have a weightable directed extension.

The terminology of lifting is motivated by the well known process, carrying the same name, which allows one to extend a domain by a minimum \perp (e.g. ).

We motivate the property of directedness from a complexity theoretic point of view, which leads to the study of the particular class of weightable directed spaces.

We recall that the weightable quasi-pseudo-metric spaces (also refered to as partial metric spaces) have been introduced in . The topological study of these spaces has been continued in and in the survey paper ``Nonsymmetric Topology'' (), where several related open problems are stated. Recently these structures also have been studied in and , as part of the development of a computational model for metric spaces.

We show that weightable directed spaces are not metrizable and analyze the weighting functions of these spaces. The weightings of a weightable directed space are shown to be upper weightings among which there is a unique ``fading'' weighting which generates all weightings of the space.

As an illustration of a class of weightable directed spaces, the study of the \sqcup-invariant join semilattices () is continued and the dual of the class of the weightable spaces is characterized in this context.

This leads to a solution of Problem 7 ``Characterize the quasi-uniformities with a countable base which are induced by a weighted quasi-metric'' of for the class of quasi-uniformly continuous join semilattices and an application is given in the context of the dimension theory for modular lattices (e.g. ).

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Date received: July 7, 1997

Copyright © 1997 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 # caao-91.