Some problems on selections for hyperspace topologies

Valentin Gutev; Tsugunori Nogura

doi:10.4995/agt.2004.1995

Authors

Valentin Gutev University of Natal
Tsugunori Nogura Ehime University

DOI:

https://doi.org/10.4995/agt.2004.1995

Keywords:

Selections, Hyperspaces, Ordered spaces, Complete metric spaces

Abstract

The theory of hyperspaces has attracted the attention of many mathematicians who have found a large variety of its applications during the last decades. The theory has taken also its natural course and has yielded lots of problems which, besides their independent inner beauty, provide ties with numerous classical fields of mathematics. In the present note we are concerned with some open problems about selections for hyperspace topologies which have been in the scope of our recent research interests.

Downloads

Download data is not yet available.

Author Biographies

Valentin Gutev, University of Natal

School of Mathematical and Statistical Sciences

Tsugunori Nogura, Ehime University

Department of Mathematics

References

G. Artico and U. Marconi, Selections and topologically well-ordered spaces, Topology Appl. 115 (2001), 299–303. http://dx.doi.org/10.1016/S0166-8641(00)00072-9

G. Artico, U. Marconi, J. Pelant, L. Rotter, and M. Tkachenko, Selections and suborderability, Fund. Math. 175 (2002), no. 1, 1–33.

G. Beer, Topologies on closed and closed convex sets, Mathematics and its applications, vol. 268, Kluwer Academic Publishers, The Netherlands, 1993.

G. Beer, A. Lechicki, S. Levi, and S. Naimpally, Distance functionals and suprema of hyperspace topologies, Ann. Mat. Pure Appl. 162 (1992), 367–381. http://dx.doi.org/10.1007/BF01760016

D. Bertacchi and C. Costantini, Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space, Topology Appl. 88 (1998), 179–197. http://dx.doi.org/10.1016/S0166-8641(97)00175-2

M. Choban, Many-valued mappings and Borel sets. I, Trans. Moscow Math. Soc. 22 (1970), 258–280.

C.Costantini and V. Gutev, Recognizing special metrics by topological properties of the “metric”-proximal hyperspace, Tsukuba J. Math. 26 (2002), no. 1, 145–169.

R. Engelking, R. W. Heath, and E. Michael, Topological well-ordering and continuous selections, Invent. Math. 6 (1968), 150–158. http://dx.doi.org/10.1007/BF01425452

S. Fujii and T. Nogura, Characterizations of compact ordinal spaces via continuous selections, Topology Appl. 91 (1999), 65–69. http://dx.doi.org/10.1016/S0166-8641(97)00243-5

S. García-Ferreira, V. Gutev, T. Nogura, M. Sanchis, and A. Tomita, Extreme selections for hyperspaces of topological spaces, Topology Appl. 122 (2002), 157–181. http://dx.doi.org/10.1016/S0166-8641(01)00141-9

S. García-Ferreira and M. Sanchis, Weak selections and pseudocompactness, preprint, 2001.

V. Gutev, Selections and hyperspace topologies via special metrics, Topology Appl. 70 (1996), 147–153. http://dx.doi.org/10.1016/0166-8641(95)00092-5

V. Gutev, Fell continuous selections and topologically well-orderable spaces II, Proceedings of the Ninth Prague Topological Symposium (2001), Topology Atlas, Toronto, 2002, pp. 157–163 (electronic).

V. Gutev and T. Nogura, Fell continuous selections and topologically well-orderable spaces, Internal Report No. 13/99, University of Natal.

V. Gutev and T. Nogura , Selections for Vietoris-like hyperspace topologies, Proc. London Math. Soc. 80 (2000), no. 3, 235–256.

V. Gutev and T. Nogura , Selections and order-like relations, Applied General Topology 2 (2001), 205–218.

V. Gutev and T. Nogura , Vietoris continuous selections and disconnectedness-like properties, Proc. Amer. Math. Soc. 129 (2001), 2809–2815. http://dx.doi.org/10.1090/S0002-9939-01-05883-X

E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. http://dx.doi.org/10.1090/S0002-9947-1951-0042109-4

J. van Mill, J. Pelant, and R. Pol, Selections that characterize topological completeness, Fund. Math. 149 (1996), 127–141.

J. van Mill and E. Wattel, Selections and orderability, Proc. Amer. Math. Soc. 83 (1981), no. 3, 601–605.

T. Nogura and D. Shakhmatov, Characterizations of intervals via continuous selections, Rendiconti del Circolo Matematico di Palermo, Serie II, 46 (1997), 317–328. http://dx.doi.org/10.1007/BF02977032

T. Nogura and D. Shakhmatov , Spaces which have finitely many continuous selections, Bollettino dell’UnioneMatematica Italiana 11-A (1997), no. 7, 723–729.