The sequence selection properties of Cp(X)

For a Tychonoff space X, we denote by Cp(X) the space of all real-valued continuous functions on X with the topology of pointwise convergence. We show the following: (1) for the closed unit interval I, Cp(I) does not have the weak sequence selection property; (2) if X is a QN-space, then Cp(X) is an α1-space. These results answer problems posed by M. Scheepers. Also we give characterizations of the α1-property, the α2-property (i.e. the sequence selection property) and the weak sequence selection property of Cp(X) in terms of covering properties of X.