Bornological convergences on local proximity spaces and ωµ −metric spaces

Abstract

The main topics of this thesis are local proximity spaces jointly with some bornological convergences naturally related to them, and ωµ −metric spaces, in particular those which are Atsuji spaces (or UC spaces), jointly with their hyperstructures. Local proximities spaces carry with them two particular features: proximity [48] and boundedness [37], [40]. Proximities allow us to deal with a concept of nearness even though not providing a metric. Proximity spaces are located between topological and metric spaces. Boundedness is a natural generalization of the metric boundedness. When trying to refer macroscopic phenomena to local structures, local proximity spaces appear as a very attractive option. For that, jointly with Prof. A. Di Concilio, in a first step we displayed a uniform procedure as an exhaustive method of generating all local proximity spaces starting from unform spaces and suitable bornologies. After that, we looked at suitable topologies for the hyperspace of a local proximity space. In contrast with the proximity case, in which there is no canonical way of equipping the hyperspaces with a uniformity, the same with a proximity, the local proximity case is simpler. Apparently, at the beginning, we have three natural different ways to topologize the hyperspace CL(X) of all closed non-empty subsets of X: we can think at a local Fell hypertopology or a kind of hit and far-miss topology or also a particular uniform bornological topology. We proved that they match. In the light of the previous local proximity results, we looked for necessary and sufficient conditions of uniform nature for two different uniform bornological convergences to match. This led us to focus on a special class of uniformities: those with a linearly ordered base. They are connected with an interesting generalization of metric spaces, ωµ −metric spaces. These spaces are endowed with special distances valued in ordered abelian additive groups. Furthermore, in relation with ωµ−metric spaces, we looked at generalizations of well known hyperspace convergences, as Hausdorff and Kuratowski convergences obtaining analogue results with respect to the standard case, [28]. Finally, we dealt with Atsuji spaces.We were interested in the problem of constructing a dense extension Y of a given topological space X, which is Atsuji and in which X is topologically embedded. When such an extension there exists, we say that the space X is Atsuji extendable. Atsuji spaces play an important role above all because they allow us to deal with a very nice structure when we concentrate on the most significant part of the space, that is the derived set. Moreover, we know that each continuous function between metric or uniform spaces is uniformly continuous on compact sets. It is possible to have an analogous property on a larger class of topological spaces, Atsuji spaces. They are situated between complete metric spaces and compact ones. We proved a necessary and sufficient condition for a metrizable spaceX to be Atsuji extendable.Moreover we looked at conditions under which a continuous function f X 􀀀 R can be continuously extended to the Atsuji extension Y of X. UC metric spaces admit a very long list of equivalent formulations. We extended many of these to the class of ωµ−metric spaces. The results are contained in [29]. Finally it is presented the idea about the work done jointly with Professor J.F. Peters ( University of Manitoba , Canada). Our research involved the study of more general proximities leading to a kind of strong farness, [52]. Strong proximities are associated with Lodato proximities and the Efremoviˇc property.We say that A and B are −strongly far, where is a Lodato proximity, and we write ~ if and only if A ~ B and there exists a subset C of X such that A ~ X C and C ~ B, that is the Efremoviˇc property holds on A and B. Related to this idea we defined also a new concept of strong nearness, [53]. Starting by these new kinds of proximities we introduced also new kinds of hit-and-miss hypertopologies, concepts of strongly proximal continuity and strong connectedness. Finally we looked at some applicaii tions that in our opinion might reveal interesting.