The sequence selection properties of C_P(X)

摘要

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.