On metric spaces with the Haver property which are Menger spaces

A metric space (X,d) has the Haver property if for each sequence ϵ1,ϵ2,… of positive numbers there exist disjoint open collections V1,V2,… of open subsets of X, with diameters of members of Vi less than ϵi and covering X, and the Menger property is a classical covering counterpart to σ-compactness. We show that, under Martin's Axiom MA, the metric square (X,d)×(X,d) of a separable metric space with the Haver property can fail this property, even if X2 is a Menger space, and that there is a separable normed linear Menger space M such that (M,d) has the Haver property for every translation invariant metric d generating the topology of M, but not for every metric generating the topology. These results answer some questions by L. Babinkostova [L. Babinkostova, When does the Haver property imply selective screenability? Topology Appl. 154 (2007) 1971–1979; L. Babinkostova, Selective screenability in topological groups, Topology Appl. 156 (1) (2008) 2–9].