On metric spaces with the Haver property which are Menger spaces

Abstract

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].

A metric space with the Haver property whose square fails this property

“…One can show that under Martin's Axiom, if the Hurewicz property is replaced here by the Menger property, the first statement is no longer true, and the second one fails even if we assume that the space has property C and its square has the Menger property; see [12] (this answers Problem 4 from [2] and shows that the answers to Problems 1 and 2 in [2] are negative, even if we assume the Menger property of the space).…”

A homeomorphism between strong measure zero sets whose graph is not of strong measure zero

“…Our base will be the following modification of Michael's Lemma 5.2 in [8], taken from [9] (cf. also [1,2,10]).…”

“…[9, Section 5]. We shall omit also a proof of the following lemma, which is essentially the part (ii) of Proposition 5.1 in [9]. We are ready now for a refinement of Theorem 1.1.…”

“…The proof of Theorem 1.1, given in Section 2, is based on some ideas from [9] (although the subject of [9] was different). As in [9], this approach combined with a method of Michael introduced in [8] provides some refinements of Theorem 1.1.…”

“…As in [9], this approach combined with a method of Michael introduced in [8] provides some refinements of Theorem 1.1.…”