QN-spaces, wQN-spaces and covering properties

The main results of the paper are as follows: covering characterizations of wQN-spaces, covering characterizations of QN-spaces and a theorem saying that C p ( X ) has the Arkhangelʹskiıˇ property ( α 1 ) provided that X is a QN-space. The latter statement solves a problem posed by M. Scheepers [M. Scheepers, C p ( X ) and Arhangelʹskiıˇʹs α i -spaces, Topology Appl. 89 (1998) 265–275] and for Tychonoff spaces was independently proved by M. Sakai [M. Sakai, The sequence selection properties of C p ( X ) , Preprint, April 25, 2006]. As the most interesting result we consider the equivalence that a normal topological space X is a wQN-space if and only if X has the property S 1 ( Γ shr , Γ ) . Moreover we show that X is a QN-space if and only if C p ( X ) has the property ( α 0 ) , and for perfectly normal spaces, if and only if X has the covering property ( β 3 ) .