Representing trees as relatively compact subsets of the first Baire class


S. Todorčević


We show that there is a scattered compact subset $K$ of the first Baire class, a Baire space $X$ and a separately continuous mapping $f:X\times K\arr{\mbb R}$ which is not continuous on any set of the form $G\times K$, where $G$ is a comeager subset of $X$. We also show that it is possible to have a scattered compact subset $K$ of the first Baire class which does have the Namioka property though its function space ${\mcal C}(K)$ fails to have an equivalent Fréchet-differentiable norm and its weak topology fails to be $\sigma$-fragmented by the norm.