Gap topologies in metric spaces

In this article we study gap topologies on the subsets of a metric space (X,d)(X,d) induced by a general family SS of nonempty subsets of X  . Given two families and two metrics not assumed to be equivalent, we give a necessary and sufficient condition for one induced upper gap topology to be contained in the other. This condition is also necessary and sufficient for containment of the two-sided gap topologies under the mild assumption that the generating families contain the singletons. Coincidence of upper gap topologies in the most important special cases is attractively reflected in the underlying structure of (X,d)(X,d). First and second countability of upper gap topologies is also characterized. This approach generalizes and unifies results in Beer et al. (1992) [12] and Costantini et al. (1993) [19] and gives rise to a noticeable family of subsets that lie between the totally bounded and the bounded subsets of X.