Diagonals of strongly separately continuous functions

Abstract

We study the diagonals g(x)=f(x,...,x) of strongly separately continuous mappings f:Xⁿ→Z, that is, mappings which for a fixed value of one variable are jointly continuous with respect to the others variables. We prove that for any n≥2, any topological space X, any strongly σ-metrizable equiconnected space (Z,λ) with a perfect stratification assigned with a mapping λ and every Baire class one mapping g:X→ℝ there exists a strongly separately continuous mapping f:Xⁿ→Z with the diagonal g. From this we obtain that for any PP-space space X and any strongly σ-metrizable equiconnected space (Z,λ) with a perfect stratification assigned with a mapping λ the diagonals of strongly separately continuous mappings f:Xⁿ→Z are exactly Baire class one mappings. Moreover, we prove that for a countably compact space X the diagonals of strongly separately continuous functions f:Xⁿ→ℝ coincide with the functions of Baire class one if and only if every system of functionally open pairwise disjoint sets in X is at most countable.