A note on star compact spaces with a $G_{\delta}$-diagonal

Yan-Kui Song

In this note we give an example of a Hausdorff, star compact space with a $G_\delta$-diagonal which is not metrizable, which answers negatively a question of van Mill, Tkachuk and Wilson (Problem 4.8 in [J. van Mill, V.V. Tkachuk, R.G. Wilson, Classes defined by stars and neighbourhood assignments, Topology Appl. 154 (2007), 2127-2134]).