Gap topologies in metric spaces (Q386842)

Let \((X,d)\) be a metric space. The power set of \(X\) is denoted by \({\mathcal P}(X)\). For any two members \(L\), \(M\) of \({\mathcal P}(X)\), the \textit{gap} between \(L\) and \(M\) is defined by NEWLINE\[NEWLINED_d(L,M) = \inf \{d(x,y): x\in L, y \in M\}.NEWLINE\]NEWLINE Notice that \(D_d(L,M)=+\infty\) if and only if either \(L\) or \(M\) is empty. The main purpose of this paper under review is to study gap topologies, by considering for a given collection \(\mathcal S\) of nonempty subsets of \((X,d)\) the weakest topology \(\text{G}_{\mathcal S, d}\) for which all gap functionals of the form NEWLINE\[NEWLINEC \mapsto D_d(S, C)\;\;\;(S \in \mathcal S)NEWLINE\]NEWLINE are continuous. We can decompose the gap topology \(\text{G}_{\mathcal S, d}\) into two halves, namely the upper gap topology \(\text{G}_{\mathcal S, d}^+\) and the lower gap topology \(\text{G}_{\mathcal S, d}^-\). The upper (resp. lower) gap topology is the weakest topology such that each gap functional is lower (resp. upper) semi-continuous on \({\mathcal P}(X)\). When \(\mathcal S\) contains all singletons, \(\text{G}_{\mathcal S, d}^-\) turns out to coincide with the classical lower Vietoris topology. Hence, the upper gap topology \(\text{G}_{\mathcal S, d}^+\) plays a more influential role in hyperspace theory.NEWLINENEWLINEGiven two families of nonempty subsets of \(X\) and two metrics not assumed to be equivalent, the authors give a necessary and sufficient condition for one induced upper gap topology to be contained in the other. Coincidence of upper gap topologies in the most important special cases is investigated. First and second countability of upper gap topologies is also characterized. This paper provides an approach that generalizes and unifies a number of earlier results of Beer and Costantini on hypertopologies generated by distance functionals.