Pairs of Hahn and separately continuous function

Kushnir Anastasia Serhiivna; Maslyuchenko Oleksandr Volodymyrovych

doi:https://doi.org/10.31861/bmj2021.01.18

Kushnir Anastasia Serhiivna ¹ , Maslyuchenko Oleksandr Volodymyrovych ^2,1

¹ Department of Mathematical Analysis, Yuriy Fedkovych Chernivtsi National University, Chernivtsi, 58000, Ukraine

² Institute of Mathematics, University of Silesia in Katowice, Katowice, 40-007, Poland

DOI: https://doi.org/10.31861/bmj2021.01.18

Keywords: pair of Hahn, semicontinuous function, separately continuous function, minimal layering, maximal layering

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Abstract

In this paper we continue the study of interconnections between separately continuous function which was started by V. K. Maslyuchenko. A pair $(g,h)$, of functions on a topological space is called a pair of Hahn if $g ≤ h, g$ is an upper semicontinuous function and $h$ is a lower semicontinuous function. We say that a pair of Hahn $(g,h)$ is generated by a function $f$, which depends on two variables, if the infimum of $f$ and the supremum of $f$ with respect to the second variable equals $g$ and $h$ respectively. We prove that for any perfectly normal space $X$ and non-pseudo compact space $Y$ every pair of Hahn on $X$ is generated by a continuous function on $X$ x $Y$. We also obtain that for any perfectly normal space $X$ and for any space $Y$ having non-scattered compactification any pair of Hahn on $X$ is generated by a separately continuous function on $X$ x $Y$.

References

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ACS Style: Kushnir, A.S.; Maslyuchenko, O.V. Pairs of Hahn and separately continuous function. Bukovinian Mathematical Journal. 2021, 9 https://doi.org/https://doi.org/10.31861/bmj2021.01.18
AMA Style: Kushnir AS, Maslyuchenko OV. Pairs of Hahn and separately continuous function. Bukovinian Mathematical Journal. 2021; 9(1). https://doi.org/https://doi.org/10.31861/bmj2021.01.18
Chicago/Turabian Style: Anastasia Serhiivna Kushnir, Oleksandr Volodymyrovych Maslyuchenko. 2021. "Pairs of Hahn and separately continuous function". Bukovinian Mathematical Journal. 9 no. 1. https://doi.org/https://doi.org/10.31861/bmj2021.01.18

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