On metric spaces with the Haver property which are Menger spaces (Q972550)

The aim of this paper is to construct topological spaces with some special properties under Martin's axiom (MA) answering some questions by \textit{L. Babinkostova} [Topology Appl. 154, No.~9, 1971--1979 (2007; Zbl 1128.54014) and ibid. 156, No.~1, 2--9 (2008; Zbl 1157.54010)]. The results: under MA, there is a separable metric space \((V,d)\) with the Haver property such that \(X\) is a \(C\)-space and Menger space, but the product space \((X,d)\times (X,d)\) does not have the Haver property. Under MA, there is a linear subspace \(M\) of a separable Hilbert space which is a Menger space and for each translation invariant metric \(d\) generating the topology of \(M\) the metric space \((M,d)\) has the Haver property and \(M\) is not a \(C\)-space. The first construction gives the following result: under MA, there exists a \(C\)-space \(X\), whose \(X^2\) has the Menger property such that the product of \(X\) with the space of the irrationals is strongly infinite-dimensional, and is not a \(C\)-space.