On Tychonoff-Type Hypertopologies

In 1975, M. M. Choban [5] introduced a new topology on the set of all closed subsets of a topological space, similar to the but weaker than it. In 1998, G. Dimov and D. Vakarelov [8] used a generalized version of this new topology, calling it . The present paper is devoted to a detailed study of Tychonoff-type topologies on an arbitrary family of subsets of a set . When contains all singletons, a description of all Tychonoff-type topologies on is given. The continuous maps of a special form between spaces of the type () are described in an isomorphism theorem. The problem of is investigated as well. Some topological properties of the hyperspaces () with Tychonoff-type topologies are briefly discussed