On Separately Continuous Functions of the First Baire Class on Squares and Mixed Products

Abstract

We consider Mykhaylyuk's problem asking whether the Baire-one property for every separately continuous real-valued function on each of the squares $X^2$ and $Y^2$ implies the same property on $X\times Y$. Assuming $\mathfrak b>ω_1$, the article obtains a negative answer using the compact cube $K=[-1,1]^{ω_1}$ and the space $E=(c_{00}(ω_1),σ(c_{00}(ω_1),\ell_\infty(ω_1)))$. It presents a finite-coordinate characterization of Baire-one functions on $E$ under this hypothesis.