On Hurewicz properties

We investigate Hurewicz properties introduced in [M. Menger, Sitzungberichte der Wiener Akademie 133 (1924) 421–444] and [W. Hurewicz, Fund. Math.  9 (1925) 193–204] and later introduced related properties of topological spaces. The main result says that for perfectly normal spaces the property mQN introduced in [L. Bukovský et al., Topology Appl. 112 (2001) 13–40] is equivalent to Hurewicz property E∗∗ω. As corollaries we obtain solution of several open problems stated in this paper. A complete overview of relationships between the considered properties is presented.