On metric spaces with the Haver property which are Menger spaces

摘要

A metric space (X,d) has the Haver property if for each sequence ∈_1,∈_2.... of positivernnumbers there exist disjoint open collections V_1,V_2.... of open subsets of X, withrndiameters of members of V_i less than ∈_i, and U_(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].