Spaces with a $\mathbb{Q}$-diagonal

Abstract

A space X has a Q-diagonal if X 2 \ ∆ has a K(Q)-directed compact cover. We show that any compact space with a Q-diagonal is metrizable, hence any Tychonoff space with a Q-diagonal is cosmic. These give positive answers to Problem 4.2 and Problem 4.8 in [3] raised by Cascales, Orihuela, and Tkachuk.