The author continues his paper [Rend. Circ. Mat. Palermo, II. Ser. 45, No. 1, 75-83 (1996; Zbl 0907.54010)], concerning hit-and-miss topologies on the set \(\text{Cl}(X)\) of nonempty closed subsets of a topological space \(X\), taking always closed sets to produce the hit-and-miss basic open sets (\(\Delta\)-topologies). Some known results on first and second countability of the hyperspaces [\textit{G. Di Maio} and \textit{L. Holá}, On hit-and-miss topologies, Rend. Accad. Sci. Fis. Mat., Napoli (4) 62, 103-124 (1995) and with \textit{E. Meccariello}, Rostocker Math. Kolloq. 52, 19-32 (1998; Zbl 0937.54009)] are generalized by dropping assertions on separation properties to weakly-\(R_0\), instead of Hausdorffness or \(T_1\). Furthermore, the author proves, that to be a Tikhonov-space, or completely regular (thus uniformizable), or \(T_3\), or regular, or to have the property \(P_{\Delta}\) with \(\Delta\) being an Uryson family, are all equivalent for the hyperspaces -- without additional requirements to the base space (theorem 2.6). Moreover, it is shown (theorem 2.8), that the following are equivalent for the hyperspaces: metrizability, pseudo-metrizability, to be second countable and regular. The general results are applied especially to Vietoris- and Fell-topology. The reviewer explicitly recommends to read the first part [loc. cit] of this very nice study on \(\Delta\)-topologies, too.