Geodesic
Vietoris-type topologies on hyperspaces
Serdica Mathematical Journal, Tome 44 (2018) no. 1-2, pp. 103-120.

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We study the (F, G)-hit-and-miss topology introduced by Clementino and Tholen [4]. The same topology was introduced independently in the preliminary version [12] of this paper under the name of Vietoris-type hypertopology (at that time we were not aware of the paper [4]). We show that the Vietoris-type hypertopology is, in general, different from the Vietoris topology. Also, some of the results of E. Michael [13] about hyperspaceswith Vietoris topology are extended to analogous results for hyperspaces with Vietoris-type topology. We obtain as well some results about hyperspaces with Vietoris-type topology which concern some problems analogous to those regarded by H.-J. Schmidt in [14].
Keywords: hyperspace, Vietoris (hyper)topology, Vietoris-type (lower-Vietoris-type;upperVietoris-type;Tychonoff-type) (hyper)topology, (F;G)-hit-and-miss topology, 54B20, 54D10, 54C05 
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Ivanova-Dimova, Elza. Vietoris-type topologies on hyperspaces. Serdica Mathematical Journal, Tome 44 (2018) no. 1-2, pp. 103-120. http://geodesic.mathdoc.fr/item/SMJ2_2018_44_1-2_a4/