<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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 α 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.</p></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 α 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.

On the Borel Classes of Set-Valued Maps of Two Variables

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"On the Borel Classes of Set-Valued Maps of Two Variables"}

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...

On the Borel Classes of Set-Valued Maps of Two Variables

Published Online: Jul 16, 2020

Page range: 81 - 95

Received: Jan 14, 2020

Accepted: Jun 28, 2020

DOI: https://doi.org/10.2478/amsil-2020-0018

Keywords
continuity, lower (upper) semicontinuous, lower (upper) quasi-continuous set-valued map, Borel classes of set-valued maps

© 2020 Ľubica Holá et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

On the Borel Classes of Set-Valued Maps of Two Variables

Published Online: Jul 16, 2020

Page range: 81 - 95

Received: Jan 14, 2020

Accepted: Jun 28, 2020

DOI: https://doi.org/10.2478/amsil-2020-0018

Keywordscontinuity, lower (upper) semicontinuous, lower (upper) quasi-continuous set-valued map, Borel classes of set-valued maps

© 2020 Ľubica Holá et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
continuity, lower (upper) semicontinuous, lower (upper) quasi-continuous set-valued map, Borel classes of set-valued maps