The bounded Vietoris topology and applications (Q1277601)

Many efforts have been done in order to describe most of the known hypertopologies in a unified way, either for studying the relationships among them, or for finding general properties which allow to choose the most suitable hypertopology for a certain application. These unified approaches have also led to the definition of new hypertopologies, which naturally fill the gaps between the old ones. This paper studies one of these new hypertopologies, the bounded Vietoris topology, which originated in [\textit{G. Beer} and the first author, Trans. Am. Math. Soc. 335, No. 2, 805-822 (1993; Zbl 0810.54011)] from the description of the hypertopologies on the closed subsets of a metric space as weak topologies generated by families of functionals defined on closed sets. Suppose \((X,d)\) is a metric space. Well-known hypertopologies are defined by the functionals \(\rho(x,\cdot)\) when \(x\) ranges in \(X\) and \(\rho\) in one of the following families: the family of all metrics that are equivalent to \(d\), the family of all metrics that are uniformly equivalent to \(d\) and the family of all metrics that are uniformly equivalent to \(d\) and that determine the same bounded sets as \(d\). The picture becomes complete if one adds the hypertopology defined by \(\rho\) ranging in the family of all metrics that are equivalent to \(d\) and that determine the same bounded sets as \(d\). This is by definition the bounded Vietoris topology, and the fact that the hypertopology obtained without the condition on the bounded sets is the Vietoris topology explains the choice of the name. This paper investigates three main aspects of the bounded Vietoris topology. In section 3 it is compared to some known hypertopologies, in section 4 it is analyzed from a topological point of view, in section 5 we present some applications in optimization.