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.

The following statement is proved to be independent from $[LB+\neg CH]$: $(*)$ Let $X$ be a Tychonoff space with $c\left(X\right)\le {\aleph }_0$ and $\pi w\left(X\right)<ℭ$. Then a union of less than $ℭ$ of nowhere dense subsets of $X$ is a union of not greater than $\pi w\left(X\right)$ of nowhere dense subsets.