Topology Atlas Document # ppae-06


On Tychonoff-type hypertopologies

Georgi Dimov, Franco Obersnel and Gino Tironi

Proceedings of the Ninth Prague Topological Symposium (2001) pp. 51-70

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.

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Copyright © 2002 Charles University and Topology Atlas. Published April 2002.