Two properties of Cp(X) weaker than the Fréchet Urysohn property

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. In this paper, we study the κ-Fréchet Urysohn property and the weak Fréchet Urysohn property of Cp(X). Our main results are that (1) Cp(X) is κ-Fréchet Urysohn iff X has property (κ) (i.e. every pairwise disjoint sequence of finite subsets of X has a strongly point-finite subsequence). In particular, if Cp(X) is a Baire space, then it is κ-Fréchet Urysohn; (2) among separable metrizable spaces, every λ-space has property (κ) and every space having property (κ) is always of the first category; (3) every analytic space has the ω-grouping property, hence for every analytic space X, Cp(X) is weakly Fréchet Urysohn.