More about Spaces with a Small Diagonal

Hušek defines a space X to have a small diagonal if each uncountable subset of X disjoint from the diagonal, has an uncountable subset whose closure is disjoint from the diagonal. Hušek proved that a compact space of weight ω1 which has a small diagonal, will be metrizable, but it remains an open problem to determine if the weight restriction is necessary. It has been shown to be consistent that each compact space with a small diagonal is metrizable, in particular, Juhasz and Szentmiklossy [JS92] proved that this holds in models of CH. In this paper we prove that it also follows from the Proper Forcing Axiom (PFA). We also present two (consistent) examples of countably compact non-metrizable spaces with small diagonal, one of which maps perfectly onto ω1.