## Bornological convergences on local proximity spaces and Ï‰µ −metric spaces

##### Abstract

The main topics of this thesis are local proximity spaces jointly with some
bornological convergences naturally related to them, and Ï‰µ −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, Ï‰µ −metric spaces. These spaces are endowed with special
distances valued in ordered abelian additive groups.
Furthermore, in relation with Ï‰µ−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 Ï‰µ−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 EfremoviË‡c 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 EfremoviË‡c 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.