On the Borel classes of set-valued maps of two variables

L'ubica Holá; Grażyna Kwiecińska

Published: 2020-07-16

Vol. 34 No. 1 (2020)

On the Borel classes of set-valued maps of two variables

L'ubica Holá , Grażyna Kwiecińska

Abstract

Using the Borel classification of set-valued maps, we present here some new results on set-valued maps which are similar to some of the well known theorems on functions due to Lebesgue and Kuratowski. We consider set-valued maps of two variables in perfectly normal topological spaces. It was proved in [11] that a set-valued map lower semicontinuous (i.e. of lower Borel class 0) in the first and upper semicontinuous (i.e. of upper Borel class 0) in the second variable is of upper Borel class 1 and also (with stronger assumptions) of lower Borel class 1. This result cannot be generalized into higher Borel classes. In this paper we show that a set-valued map of the upper (resp. lower) Borel class in the first and lower semicontinuous and upper quasicontinuous (upper semicontinuous and lower quasicontinuous) in the second variable is of the lower (resp. upper) Borel class α+1. Also other cases are considered.

Keywords:

continuity , lower (upper) semicontinuous , lower (upper) quasicontinuous set-valued map , Borel classes of set-valued maps

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Holá, L., & Kwiecińska, G. (2020). On the Borel classes of set-valued maps of two variables. Annales Mathematicae Silesianae, 34(1), 81–95. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13634

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L'ubica Holá

Affiliation:

Institute of Mathematics, Academy of Sciences, Slovakia Slovakia

Grażyna Kwiecińska

gkk@mat.ug.edu.pl
https://orcid.org/0000-0002-8913-1941

Affiliation:

Instytut Matematyki, Uniwersytet Gdański Poland

Vol. 34 No. 1 (2020)
Published: 2020-07-20

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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