Email this article ON CONVERGENCE IN THE SPACE OF CLOSED SUBSETS OF A METRIC SPACE
- Authors: Panasenko E.A.1
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Affiliations:
- Tambov State University named after G.R. Derzhavin
- Issue: Vol 22, No 3 (2017)
- Pages: 565-570
- Section: Articles
- URL: https://journal-vniispk.ru/2686-9667/article/view/362844
- DOI: https://doi.org/10.20310/1810-0198-2017-22-3-565-570
- ID: 362844
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Abstract
We consider the space closX of closed subsets of unbounded (not necessarily separable) metric space X, ϱ X endowed with the metric ρ X cl introduced in [ Zhukovskiy E.S., Panasenko E.A. // Fixed Point Theory and Applications. 2013:10]. It is shown that if any closed ball in the space X, ϱ X is totaly bounded, then convergence in the space clos X , ρ X cl of a sequence F i i=1 ∞ to F is equivalent to convergence in the sense of Wijsman, that is to convergence for each x∈X of the distances ϱ X x, F i to ϱ X x, F .
About the authors
Elena Aleksandrovna Panasenko
Tambov State University named after G.R. Derzhavin
Email: panlena_t@mail.ru
Candidate of Physics and Mathematics, Associate Professor of the Functional Analysis Department Tambov, the Russian Federation
References
Lechicki A., Levi S. Wijsman convergence in the hyperspace of a metric space // Bollettino U.M.I. 1987. V. 7. P. 439-451. Francaviglia S., Lechicki A., Levi S. Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions // J. Math. Anal. Appl. 1985. V. 112. n. 2. P. 347-370. Wijsman R.A. Convergence of sequences of convex sets, cones and functions. II. Trans. Amer. Math. Soc. 1966. V. 123. P. 32-45. Beer G. Metric spaces with nice closed balls and distance functions for closed sets // Bull. Austral. Math. Soc. 1987. V. 35. P. 81-96. Zhukovskiy E.S., Panasenko E.A. On multi-valued maps with images in the space of closed subsets of a metric space // Fixed Point Theory and Applications. 2013. 2013:10 doi: 10.1186/1687-1812-2013-10. Жуковский Е.С., Панасенко Е.А. Определение метрики пространства clos∅(X) замкнутых подмножеств метрического пространства X и свойства отображений со значениями в clos∅(Rn) // Математический сборник. 2014. Т. 205. № 9. C. 65-96.
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