A metric space (X, d) has the Haver property if for each sequence 1 , 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 i and ∞ i=1 V i 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 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].

No abstract

) fails this property. In particular, the square of a separable complete metric space with the Haver property may fail this property. Our results answer some questions posed by Babinkostova in 2007.