On metric spaces with the Haver property which are Menger spaces

A metric space (X, d) has the Haver property if for each sequence epsilon(1), epsilon(2), of positive numbers there exist disjoint open collections V-1, V-2. of open subsets of X, with diameters of members of V-1 less than epsilon(1) and boolean OR(infinity)(i=1) V-1 covering X. and the Meager property is a classical covering counterpart to sigma-compactness We show that. under Maitm's Axiom MA. the metric square (X, d) x (X, d) of a separable metric space with the Haver property can fail this property, even if X-2 is a Meager space. and that there is a separable normed linear Meager space M such that (A4 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] (C) 2009 Elsevier B V All rights reserved