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A UNIFIED APPROACH TO CONTINUOUS, MEASURABLE SELECTIONS, AND SELECTIONS FOR HYPERSPACES

Part of: Set theory Classical measure theory Maps and general types of spaces defined by maps Set functions, measures and integrals with values in abstract spaces Basic constructions

Published online by Cambridge University Press:  21 May 2018

Fotis H. Mavridis
Fotis H. Mavridis*
Affiliation:
National Technical University of Athens, Department of Mathematics, Zografou Campus, 15780 Athens, Greece email [email protected]
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Abstract

In this paper we provide a unified approach, based on methods of descriptive set theory, for proving some classical selection theorems. Among them is the zero-dimensional Michael selection theorem, the Kuratowski–Ryll-Nardzewski selection theorem, as well as a known selection theorem for hyperspaces.

MSC classification

Primary: 54C65: Selections 54C60: Set-valued maps 54B20: Hyperspaces 28B20: Set-valued set functions and measures; integration of set-valued functions; measurable selections
Secondary: 03E15: Descriptive set theory 28A05: Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets
Type
Research Article
Information
Mathematika , Volume 64 , Issue 3 , 2018 , pp. 607 - 627
Copyright
Copyright © University College London 2018 

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Footnotes

This paper contains a part of the author’s doctoral thesis written under the supervision of Professor Alexander D. Arvanitakis at the National Technical University of Athens. The author would like to thank Professor Alexander Arvanitakis, Professor Apostolos Giannopoulos, Dr Elena Papanikolaou, and the referee for their valuable feedback.

References

Aliprantis, Ch. D. and Border, K., Infinite Dimensional Analysis, Springer (Berlin, Heidelberg, 2006).Google Scholar
Argyros, S. A. and Arvanitakis, A., A characterization of regular averaging operators and its consequences. Studia Math. 151 2002, 207–226.Google Scholar
Arvanitakis, A., A simultaneous selection theorem. Fund. Math. 219 2012, 1–14.Google Scholar
Choban, M. M., Many-valued mappings and Borel sets I. Trans. Moscow Math. Soc. 22 1970, 258–280.Google Scholar
Denkowski, Z., Migórski, S. and Papageorgiou, N. S., An Introduction to Nonlinear Analysis: Theory, Springer (New York, 2003).Google Scholar
Ditor, S., Averaging operators in C(S) and lower semicontinuous sections of continuous maps. Trans. Amer. Math. Soc. 175 1973, 195–208.Google Scholar
Ditor, S. and Haydon, R., On absolute retracts, P(S), and complemented subspaces of C(D 𝜔1 ). Studia Math. 56 1976, 243–251.Google Scholar
Engelking, R., General Topology, PWN (Warszawa, 1977).Google Scholar
Engelking, R., Heath, R. W. and Michael, E., Topological well-ordering and continuous selections. Invent. Math. 6 1968, 150–158.Google Scholar
Gutev, V., Completeness, sections and selections. Set-Valued Anal. 15 2007, 275–295.Google Scholar
Gutev, V., Selections and hyperspaces. In Recent Progress in General Topology III (eds Hart, K. P., van Mill, J. and Simon, P.), Atlantis Press (2014), 535–579.Google Scholar
Gutev, V. and Nogura, T., Selection problems for hyperspaces. In Open Problems in Topology II (ed. Pearl, E.), Elsevier (Amsterdam, 2007), 161–170.Google Scholar
Haydon, R., On a problem of Pelczyński: Milutin spaces, Dugundji spaces and AE(0-dim). Studia Math. 52 1974, 23–31.Google Scholar
Haydon, R., Embedding D 𝜏 in Dugundji spaces, with an application to linear topological classification of spaces of continuous functions. Studia Math. 56 1976, 229–242.Google Scholar
Hu, S. and Papageorgiou, N. S., Handbook of Multivalued Analysis, Volume I: Theory, Kluwer (Dordrecht, 1997).Google Scholar
Kechris, A., Classical Descriptive Set Theory, Springer (New York, 1995).Google Scholar
Kuratowski, K. and Ryll-Nardzewski, C., A general theorem on selectors. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 1965, 397–403.Google Scholar
Mägerl, G., A unified approach to measurable and continuous selections. Trans. Amer. Math. Soc. 245 1978, 443–452.Google Scholar
Maitra, A. and Rao, B. V., Selection theorems and the reduction principle. Trans. Amer. Math. Soc. 202 1975, 57–66.Google Scholar
Michael, E., Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71 1951, 152–182.Google Scholar
Michael, E., Continuous selections I. Ann. of Math. (2) 63 1956, 361–382.Google Scholar
Michael, E., Selected selection theorems. Amer. Math. Monthly 63 1956, 233–238.Google Scholar
Michael, E., A theorem on semi-continuous set-valued functions. Duke Math. J. 26 1959, 647–651.Google Scholar
Michael, E. and Pixley, C., A unified theorem on continuous selections. Pacific J. Math. 87 1980, 187–188.Google Scholar
Repovs̆, D. and Semenov, P. V., Continuous Selections of Multivalued Mappings, Kluwer (Dordrecht, 1998).Google Scholar
Repovs̆, D. and Semenov, P. V., Continuous selections of multivalued mappings. In Recent Progress in General Topology II (eds Hušek, M. and van Mill, J.), North Holland (Amsterdam, 2002), 423–461.Google Scholar
Valov, V., On a theorem of Arvanitakis. Mathematika 59 2013, 250–256.Google Scholar
Willard, S., General Topology, Dover (New York, 2004) (republication).Google Scholar
Yamauchi, T., On a simultaneous selection theorem. Studia Math. 215 2013, 1–9.Google Scholar