A.A. Tolstonogov. Space of continuous set-valued mappings with closed unbounded values ... P. 200-208

We consider a space of continuous multivalued mappings defined on a locally compact space ${\cal T}$ with countable base. Values of these mappings are closed not necessarily bounded sets from a metric space $(X,d(\cdot))$ in which closed balls are compact. The space $(X,d(\cdot))$ is locally compact and separable. Let $Y$ be a dense countable set from $X$. The distance $\rho(A,B)$ between sets $A$ and $B$ from the family $CL(X)$ of all nonempty closed subsets of $X$ is defined as
$$\rho(A,B)=\sum_{i=1}^\infty \frac{1}{2^i}\,\frac{\mid d(y_i,A)-d(y_i,B)\mid}{1+\mid d(y_i,A)-d(y_i,B)\mid},$$
where $d(y_i,A)$ is the distance from a point $y_i \in Y$ to the set $A$. This distance is independent of the choice of the set $Y$, and the function $\rho(A,B)$ is a metric on the space $CL(X)$. The convergence of a sequence of sets $A_n$, $n\ge 1$, from the metric space $(CL(X),\rho(\cdot))$ is equivalent to the Kuratowski convergence of this sequence. We prove the completeness and separability of the space $(CL(X),\rho (\cdot))$ and give necessary and sufficient conditions for the compactness of sets in this space. The space $C({\cal T}, CL(X))$ of all continuous mappings from ${\cal T}$ to $(CL(X),\rho (\cdot))$ is endowed with the topology of uniform convergence on compact sets from ${\cal T}$. We prove the completeness and separability of the space $C({\cal T}, CL(X))$ and give necessary and sufficient conditions for the compactness of sets in this space. These results are reformulated for the space $C(T,CCL(X))$, where $T=[0,1]$, $X$ is a finite-dimensional Euclidean space, and $CCL(X)$ is the space of all nonempty closed convex sets from $X$ with the metric $\rho(\cdot)$. This space plays a crucial role in the study of sweeping processes. A counterexample showing the significance of the assumption of the compactness of closed balls from $X$ is given.

Keywords: unbounded sets, Kuratowski convergence, compactness.

The paper was received by the Editorial Office on September 25, 2017.

Aleksandr Aleksandrovich Tolstonogov, RAS Corresponding Member, Dr. Phys.-Math. Sci., Prof.,
Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy
of Sciences, Irkutsk, 664033 Russia, e-mail: aatol@icc.ru .


1. Tolstonogov A.A. Investigation of a new class of control systems. Dokl. Math., 2012, vol. 85, no. 2, pp. 178–180. doi: 10.1134/S1064562412020056 .

2. Tolstonogov A.A. Control sweeping processes. J. Convex Analysis, 2016, vol. 23, no. 4, pp. 1099–1123.

3. Shouchuan Hu, Papageorgiou N.S. Handbook of multivalued analysis. Theory. Vol. 1. Ser. Math. Its Appl., vol. 149, Dordrecht, Boston, London: Kluwer, 1997, 964 p. ISBN: 0792346823 .

4. Panasenko E.A., Rodina L.I., Tonkov E.L. The space clcv($R^n$) with the Hausdorff-Bebutov metric and differential inclusions. Proc. Steklov Inst. Math. (Suppl.), 2011, vol. 275, suppl. 1, pp. 121–136. doi: 10.1134/S0081543811090094 .

5. Zhukovskiy E.S., Panasenko E.A. On multivalued maps with images in the space of closed subset of a metric space. Fixed Point Theory. Appl., 2013, no. 10, 21 p. doi: 10.1186/1687-1812-2013-10 .

6. Tolstonogov A.A. Compactness in the space of set-valued mappings with closed values. Dokl. Math., 2014, vol. 89, no. 3, pp. 293–295. doi: 10.1134/S1064562414030120 .

7. Bourbaki N. $\acute E$l$\acute e$ments de Math$\acute e$matique, Premi$\acute e$r partie, Livre III, volume Topologie G$\acute e$n$\acute e$rale. Paris: Hermann, 1960, 366 p. ISBN: 2903684002X. Translated to Russian under the title Obshchaya topologiya. Ispol’zovanie veshchestvennykh chisel v obshchei topologii. Funktsional’nye prostranstva. Svodka rezul’tatov. Moscow, Nauka Publ., 1975, 408 p.

8. Kuratowski K. Topology. Vol. I. N Y, London: Acad. Press, 1966, 560 p. ISBN: 978-0-12-429201-7 . Translated to Russian under the title Topologiya. T. 1. Moscow, Mir Publ., 1966, 594 p.

9. Beer G. Metric spaces with nice closed balls and distance functions for closed sets. Bull. Australian Math. Soc., 1987, vol. 35, no. 1, pp. 81–96. doi: 10.1017/S000497270001306X.

10. Bourbaki N. El$\acute e$ments de math$\acute e$matique, Fascicule II, Livre III, Topologie g$\acute e$n$\acute e$rale, Chap. 1, Structures topologiques, Chap. 2, structures uniformes. Paris: Hermann, 1965, 255 p. ISBN(1971 ed.): 3-540-33936-1 . Translated to Russian under the title Obshchaya topologiya. Osnovnye struktury. Moscow, Nauka Publ., 1968, 275 p.

11. Kuratowski K. Topology. Vol. II. N Y, London: Acad. Press, 1968, 608 p. ISBN: 978-0-12-429202-4 . Translated to Russian under the title Topologiya. T. 2. Moscow, Mir Publ., 1969, 624 p.

12. Beer G. On convergence of closed sets in a metric space and distance functions. Bull. Australian Math. Soc., 1985, vol. 31, pp. 421–432. doi: 10.1017/S0004972700009370 .

13. Matheron G. Random sets and integral geometry. New York: Wiley, 1975, 261 p. ISBN: 978-0-471-57621-1 . Translated to Russian under the title Sluchainye mnozhestva i integral’naya geometriya. Moscow, Mir Publ., 1978, 318 p.