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There is an open problem in this paper:

J. van Mill, V.V. Tkachuk, R.G. Wilson, "Classes defined by stars and neighbourhood assignments", Topology and its Applications, Vol. 154, Issue 10, 2007, pp. 2127–2134.

Problem 4.8. Is a regular star compact space metrizable if it has a $G_\delta$-diagonal?

A topological space $X$ is said to be star compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a compact subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$. More on star compactness see here.

My question is this: Is always the cardinality of such regular star compact space less than $2^{\omega_0}$?

Thanks for any help. Any help to solve such open problem is very welcome.

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