On the Borel classes of set-valued maps of two variables
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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Institute of Mathematics, Academy of Sciences, Slovakia Slovakia
Instytut Matematyki, Uniwersytet Gdański Poland
https://orcid.org/0000-0002-8913-1941
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