On hereditarily Baire spaces, σ-fragmentability of mappings and Namioka property

Let X=∏s∈SXs be the product of metrizable spaces. We prove that if the spaces Xs are hereditarily Baire, then X is Baire, and moreover, X has the Namioka property: any map f:X→C(K) into the function space which is pointwise continuous, has a point of continuity with respect to the norm in C(K). In case of arbitrary metrizable factors, we show that any pointwise continuous f:X→C(K) is σ-fragmented by the norm, extending a theorem from [Jayne et al., J. Funct. Analysis 117 (1993) 243-273] to the product spaces.