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