  A space $X$ has a $\mathbb{Q}$-diagonal if $X^2\setminus \Delta$ has a
$\mathcal{K}(\mathbb{Q})$-directed compact cover. We show that any compact
space with a $\mathbb{Q}$-diagonal is metrizable, hence any Tychonorff space
with a $\mathbb{Q}$-diagonal is cosmic. These give a positive answer to Problem
4.2 and Problem 4.8 in \cite{COT11} raised by Cascales, Orihuela and Tkachuk.
