Character and tightness of hyperspaces with the Fell topology

For a Hausdorff space X, we denote by 2X the collection of all closed subsets of X. The Fell topology τf on 2X has as a subbase all sets of the form V− = {F ϵ 2X: F ∩ V ≠ Ø}, where V is an open subset of X and of the form (Kc)+ = {F ϵ 2X: F ∩ K = Ø}, where K is a compact subset of X. In this paper we prove that max{dc(X), klo(X), t(X)} ⩽ t(〈2X, τF〉) ⩽ max{dc(X), klo(X), χ(X)} and χ(〈2X, τF〉) = max{dc(X), kko(X), χ(X)}, where t(X) and χ(X) denote the tightness and character of X, respectively, dc(X) is the smallest cardinal number τ such that the density of every closed subset of X is less than or equal to τ; and klo(X) and kko(X) are two cardinal invariants related to the family of all compact subsets of X. We also obtain that for a locally compact space X the tightness of 〈2X, τF〉 and the character of 〈2X, τF〉 coincide. We construct a space Y such that t(〈2Y, τF〉) ≠ max{dc(Y), klo(Y), χ(Y)}. We also give an answer to a question of Beer (1993).