Assuming OCA, we shall prove that for some pairs of Fréchet ${\alpha }_4$-spaces $X,Y$, the Fréchetness 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.

We introduce the notion of a strongly Whyburn space, and show that a space $X$ is strongly Whyburn if and only if $X\times (\omega +1)$ is Whyburn. We also show that if $X\times Y$ is Whyburn for any Whyburn space $Y$, then $X$ is discrete.