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

Let X = ∏ s ∈ S X s be the product of metrizable spaces. We prove that if the spaces X s 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.