Given a Hausdorff space X, we calculate the tightness and the character of the hyperspace CL∅(X) of X, endowed with either the co-compact or the lower Vietoris topology, and give some estimates for the tightness of CL∅(X), endowed with the Fell topology.\ud \ud Some properties related to first-countability and countable tightness, such as sequentiality, Fréchet property and, less directly, radiality and pseudoradiality, are investigated as well.\ud \ud To carry out our investigation, we also consider on the base space X several cardinal functions, and we compare some of them (which are newly defined or not so well known) with other classical ones, obtaining results and counterexamples which may be of some independent interest

In this article we study gap topologies on the subsets of a metric space (X, d) induced by a general family S 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 sufficent 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). First and second countability of upper gap topologies is also characterized. This approach generalizes and unifies results in [12] and [19] and gives rise to a noticeable family of subsets that lie between the totally bounded and the bounded subsets of X.

Generalizing a result of G Beer and a result of E. Elfros, we show that if (X, d) is a separable and completely metrizable metric space, then the hyperspace of X endowed with the Wijsman topology is separable and completely metrizable. compatible metric d on X ([Bal], Theorems III.2.i and II.8, or [Ba2]).

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We study infima of families of topologies on the hyperspace of a metrizable space. We prove that Kuratowski convergence is the infimum, in the lattice of convergences, of all Wijsman topologies and that the cocompact topology on a metric space which is complete for a metric d is the infimum of the upper Wijsman topologies arising from metrics that are uniformly equivalent to d.

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