Some Results on Hyperspace Topologies

Abstract

There have been world of good achievements in hyperspace topologies. Since the start of last century, some hyperspace topologies have been introduced and developed. Speci cally, Hausdor metric and Veitoris topologies [1, 22]. The mentioned topologies are impeccable, in the sense of their usage at the least. It is a monumental belief that the most imperative hyperspace topologies have risen as topologies determined by families of geometric set functionals refer to [4]. In fact, hyperspace topologies and related set convergence notation have been considered at the outset of last century, the way we consider to the subject re ect ultimate modren contribution by mathematicians whose mandatory research interests exist outside the general topology. The revival of the subject comes from work of R. Wijsman [25] in the mid of 1960's, and its advancements over the next fteen years was to a gigantic breakthrough in the hands of U. Mosco, R. Wets, H. Attouch, and their associates. This new approach was advanced for the most part in North America, Europe, Italy and France, in particular. This monumental interest is due to fruitfulness of these various areas of application (such as probability, statistics or variational problem, for example). It also describes the e ort in comprehending their structure, common feature and general pattern in order to nd a common description for them. About this latter view, we refer the papers [4, 23, 24], devoted to a description and classi cation of the hyperspace topologies as the outset topology, namely as the weakest topologies which makes continuous families of real-valued functionals de ned on nonempty closed subsets of Y . Not only this is helpful in order to have a common description of the hyperspace topologies, but also enables us to tackle some application in an orthodox way (see [5] and [21]). Three types of hyperspace topologies which comprises familiar topologies are as follows: the hit-and-miss, the proximal hit-and-miss [1, 6, 7] and the weak topologies generated by gap and excess functionals on nonempty closed subsets of Y [1, 4, 15], respectively. As a prototype of weak hyperspace topologies, we must recognize the Wijsman topology, which is the weak topology determined by the distance functionals seen as functionals of set argument. It is a basic tool in the construction of the lattice of hyperspace topologies, above mentioned and many other familiar hyperspace topologies has risen as supreme and in ma, respectively of appropriate Wijsman topologies [3]. Let (Y; ) be an arbitrary Hausdor topological space. We denote the collection of nonempty, closed subset of Y by C(Y ). We will investigate topologies on C(Y ) such topologies are called hyperspace topologies. The focus of the thesis is to explain the hyperspace topologies. This thesis is divided into three chapters. In chapter 1, we will discuss the Wijsman and Hausdor metric topologies. Chapter 1 also helps to explain the normality of the Wijsman topology. The chapter 2 deals with the hit-and-miss and the proximal hit-and-miss topologies. The most of the known Fell, Vietoris, proximal and ball proximal topologies are discussed. In chapter 2, the normality of the Fell and Vietoris topologies is also discussed. The last chapter describes the relationship among hyperspace topologies.