Tightness, character and related properties of hyperspace topologies

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. Some properties related to first-countability and countable tightness, such as sequentiality, Fréchet property and, less directly, radiality and pseudoradiality, are investigated as well. 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.