哈恩函数对和单独的连续函数

O. Maslyuchenko, A. Kushnir
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引用次数: 0

摘要

在本文中,我们继续研究由V. K. Maslyuchenko开始的独立连续函数之间的相互关系。如果g≤h,且g是上半连续函数,h是下半连续函数,则拓扑空间上的函数对(g, h)称为Hahn对。我们说一对Hahn (g, h)是由一个函数f生成的,它依赖于两个变量,如果f对第二个变量的最小值和最大值分别等于g和h。证明了对于任意完全正规空间X和非伪紧空间Y, X上的每一对哈恩函数都是由X X Y上的一个连续函数生成的。我们还得到了对于任何完全正规空间X和具有非分散紧化的空间Y, X上的任何哈恩函数对都是由X X Y上的一个单独的连续函数生成的。
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PAIRS OF HAHN AND SEPARATELY CONTINUOUS FUNCTION
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-pseudocompact 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 .
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