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.
Mathematics Subject Classification. 54B20 54B05 (54B30 54D10 54G99).
Keywords. Tychonoff topology, Tychonoff-type topology, T-space,
commutative space, $\mathcal{O}$-commutative space,
$\mathcal{M}$-cover, $\mathcal{M}$-closed family, $P_\infty$-space.
Comments. This article is in final form.