Abstract:Assuming OCA, we shall prove that for some pairs of Fr\'echet $\alpha _4$-spaces $X, Y$, the Fr\'echetness of the product $X\times Y$ implies that $X\times Y$ is $\alpha _4$. Assuming MA, we shall construct a pair of spaces satisfying the assumptions of the theorem.
Keywords: Fr\'echet space, $\alpha _4$-space, Fr\'echet fan, $(\kappa ,\kappa )$-good set
AMS Subject Classification: Primary 54D55; Secondary 54B10, 54G20, 03E50, 03E65