In 1975, M. M. Choban \cite{C} 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 \cite{DV} 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 $\MM$ of subsets of a set $X$. When $\MM$ contains all singletons, a description of all Tychonoff-type topologies $\OO$ on $\MM$ is given. The continuous maps of a special form between spaces of the type $(\MM,\OO)$ are described in an isomorphism theorem. The problem of {\em commutability between hyperspaces and subspaces with respect to a Tychonoff-type topology} is investigated as well. Some topological properties of the hyperspaces $(\MM,\OO)$ with Tychonoff-type topologies $\OO$ are briefly discussed.
On Tychonoff-type Hypertopologies.
OBERSNEL, Franco;
2002-01-01
Abstract
In 1975, M. M. Choban \cite{C} 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 \cite{DV} 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 $\MM$ of subsets of a set $X$. When $\MM$ contains all singletons, a description of all Tychonoff-type topologies $\OO$ on $\MM$ is given. The continuous maps of a special form between spaces of the type $(\MM,\OO)$ are described in an isomorphism theorem. The problem of {\em commutability between hyperspaces and subspaces with respect to a Tychonoff-type topology} is investigated as well. Some topological properties of the hyperspaces $(\MM,\OO)$ with Tychonoff-type topologies $\OO$ are briefly discussed.Pubblicazioni consigliate
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