van Mill et al. posed in ‘Classes defined by stars and neighborhood assignments’, Topology Appl.154 (2007), 2127–2134 the following question: Is a star-compact space metrizable if it has a Gδ-diagonal? In this paper, we give a negative answer to this question.

In this paper, we present some new metrization theorems in terms of Heath g–functions.

The notion of pointwise cleavability is introduced. We clarify those results concerning cleavability which can be or can not be generalized to the case of pointwise cleavability.

The importance of compactness in this theory is shown. Among other things we prove that t, ts, πx, the property to be Fréchet-Urysohn, radiality, biradiality, bisequentiality and so on are preserved by pointwise cleavability on the class of compact Hausdorff spaces.

If X is a topological space then exp X denotes the space of non-empty closed subsets of X with the Vietoris topology and λX denotes the superextension of X Using Martin's axiom together with the negation of the continuum hypothesis the following is proved: If every discrete subset of exp X is countable the X is compact and metrizable. As a corollary, if λX contains no uncountable discrete subsets then X is compact and metrizable. A similar argument establishes the metrizability of any compact space X whose square X × X contains no uncountable discrete subsets.

It is shown that the diagonal of X has a countable neighborhood base in X × X if and only if X is a metrizable space whose set of non-isolated points is compact.