Title
The Vietoris hyperspace structure for approach spaces The Vietoris hyperspace structure for approach spaces
Author
Faculty/Department
Faculty of Sciences. Mathematics and Computer Science
Publication type
article
Publication
Budapest ,
Subject
Mathematics
Source (journal)
Acta mathematica Hungarica. - Budapest
Volume/pages
139(2013) :3 , p. 286-302
ISSN
0236-5294
ISI
000317969300008
Carrier
E
Target language
English (eng)
Full text (Publishers DOI)
Affiliation
University of Antwerp
Abstract
We show that there exists a natural approach version of the topological Vietoris hyperspace construction [16], [17] which has several advantages over the topological version. In the first place the important structural fact that the Vietoris construction can now also be considered, not only for topological but also intrinsically for metric spaces, but equally important in the second place the fact that we can considerably strengthen fundamental classic results. In this paper we mainly pay attention to properties concerning or involving compactness. As main results, in the first place we prove that it is not merely compactness of the Vietoris hyperspace which is equivalent to compactness of the original space [3] but that actually in the approach setting the indices of compactness [7], [8], [9], [10] numerically completely coincide. In the second place the well-known result [3], [4], [15] which says that if the original space is compact metric then the Vietoris topology is metrizable by the Hausdorff metric gets strengthened in the sense that in the approach setting under the same conditions the Vietoris approach structure actually coincides with the Hausdorff metric. Classic results follow as easy corollaries. Besides these main results we also draw attention to the good functorial relationship between the Vietoris approach structures and the associated topologies.
E-info
https://repository.uantwerpen.be/docman/iruaauth/4ef147/4ded501e3d5.pdf
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