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.