On metric spaces with the Haver property which are Menger spaces.
by
Elzbieta Pol
University of Warsaw
Coauthors: Roman Pol

A metric space (X, d) has the Haver property, if for each sequence r(1), r(2), ... of positive numbers there are disjoint open collections V(1), V(2), ... in X such that the diameters of members of V(i) are less than r(i) and the union of all families V(i) covers X. A metrizable space X has the property C, if for every metric d on X generating the topology, (X, d) has the Haver property. For s-compact spaces X, the Haver property of (X, d) implies the property C of X and the Haver property of any product
(X, d)×(Y, e) by a metric space (Y, e) with the Haver property. L.Babinkostova (Top. and Appl. 154 (2007)) proved that s-compactness can be replaced by the Hurewicz property. Answering a question by Babinkostova we shall show that under Martin's Axiom, the Hurewicz property can not be weakened to the Menger property - another classical covering counterpart to s-compactness. Some other related topics will be discussed.

Date received: March 18, 2008