Approximation and Baire classification of separately continuous functions on products of generalized ordered and compact spaces

Abstract

We deal with the problem of finding sufficient and necessary conditions on a generalized ordered space X under which for every compact space Y and for every separately continuous function f:X×Y→R: (a) f is of the first Baire class, i.e. X is a Moran space; (b) the function f is a pointwise limit of a sequence of continuous functions which is uniformly convergent to f with respect to each variable at every point of X×Y. We introduce PC- and wPC-properties of topological spaces and prove that wPC-property is equivalent to the perfectness in the class of GO-spaces, and every completely regular Moran Baire space has the wPC-property. We get a certain solution of the problem (b). In particular, we obtain that (b) is true for a perfect strongly zero-dimensional hereditarily Baire generalized ordered space X with Gδ-diagonal. Moreover, under the assumption of the existence of the Souslin continuum (¬SH) we construct a Baire non-measurable separately continuous function f:X×Y→R defined on the product of a perfect linearly ordered paracompact space X and a compact space Y.