The sequence selection properties of

For a Tychonoff space X, we denote by C p ( 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 , C p ( I ) does not have the weak sequence selection property; (2) if X is a QN-space, then C p ( X ) is an α 1 -space. These results answer problems posed by M. Scheepers. Also we give characterizations of the α 1 -property, the α 2 -property (i.e. the sequence selection property) and the weak sequence selection property of C p ( X ) in terms of covering properties of X.