Supernearness, a common concept of supertopologies and nearness
by
Dieter Leseberg
TU Braunschweig + FU Berlin

SUPERNERNESS, a common concept of

supertopologies and nearness

This paper is dedicated in honour of Prof. Csaszar, who always supported my mathematical work.

In 1964 Doitchinov introduced the notion of supertopological spaces in order to construct a unified theory of topological spaces, proximity spaces and uniform spaces. In 1973 Herrlich introduced nearness spaces and since that time these spaces have been used for different purposes by topologists. Some important applications of nearness spaces within topology are those of unification and extension. Moreover it has also been of interest to study questions about extensions of topological spaces and its relationships to some special classes of supertopologies and nearnesses respectively. So, Doitchinov shows that the compactly determined Hausdorff-exten- sions of a given topological space X are closely connected with a class of supertopologies on X which he called b-supertopologies. Bentley shows that those nearness spaces which can be extended to a topological space have a neat internal characterization, namely the condition is that every nearness collection should be a subset of some bunch. In this paper we present a common unification of these two mentioned concepts by using the so called supernear operators, and moreover we are going to characterize those supernearness spaces which can be extended to topological ones.

Date received: June 16, 1998