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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 2018, Volume 104, Issue 1, Pages 99–117 (Mi mz11700)

On the Metric Space of Closed Subsets of a Metric Space and Set-Valued Maps with Closed Images

E. A. Panasenko

Tambov State University named after G.R. Derzhavin

Abstract: The space $\operatorname{clos}(X)$ of all nonempty closed subsets of an unbounded metric space $X$ is considered. The space $\operatorname{clos}(X)$ is endowed with a metric in which a sequence of closed sets converges if and only if the distances from these sets to a fixed point $\theta$ are bounded and, for any $r$, the sequence of the unions of the given sets with the exterior balls of radius $r$ centered at $\theta$ converges in the Hausdorff metric. The metric on $\operatorname{clos}(X)$ thus defined is not equivalent to the Hausdorff metric, whatever the initial metric space $X$. Conditions for a set to be closed, totally bounded, or compact in $\operatorname{clos}(X)$ are obtained; criteria for the bounded compactness and separability of $\operatorname{clos}(X)$ are given. The space of continuous maps from a compact space to $\operatorname{clos}(X)$ is considered; conditions for a set to be totally bounded in this space are found.

Keywords: space of nonempty closed subsets of a metric space, total boundedness, set-valued map.

 Funding Agency Grant Number Russian Foundation for Basic Research 17-01-00553 Ministry of Education and Science of the Russian Federation 3.8515.2017/8.9 This work was performed in the framework of the base part of the Government order of the Ministry of Education and Science of the Russian Federation (grant no. 3.8515.2017/8.9) with the support of the Russian Foundation for Basic Research (grant no. 17-01-00553).

DOI: https://doi.org/10.4213/mzm11700

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English version:
Mathematical Notes, 2018, 104:1, 96–110

Bibliographic databases:

UDC: 515.124+515.126.83
Revised: 14.09.2017

Citation: E. A. Panasenko, “On the Metric Space of Closed Subsets of a Metric Space and Set-Valued Maps with Closed Images”, Mat. Zametki, 104:1 (2018), 99–117; Math. Notes, 104:1 (2018), 96–110

Citation in format AMSBIB
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