Two properties of weaker than the Fréchet Urysohn property

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. In this paper, we study the κ-Fréchet Urysohn property and the weak Fréchet Urysohn property of C p ( X ) . Our main results are that (1) C p ( 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 C p ( 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, C p ( X ) is weakly Fréchet Urysohn.