On metric spaces with the Haver property which are Menger spaces

A metric space ( X , d ) has the Haver property if for each sequence ϵ 1 , ϵ 2 , … of positive numbers there exist disjoint open collections V 1 , V 2 , … of open subsets of X, with diameters of members of V i less than ϵ i and ⋃ i = 1 ∞ V i covering X, and the Menger property is a classical covering counterpart to σ-compactness. We show that, under Martinʹs Axiom MA, the metric square ( X , d ) × ( X , d ) of a separable metric space with the Haver property can fail this property, even if X 2 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].