Function spaces with a countable -network at a point

For a Tychonoff space X, we denote by C p ( X ) ( C k ( X ) ) the space of all real-valued continuous functions on X with the topology of pointwise convergence (the compact-open topology). In this paper, we show that C p ( X ) has a countable c s ∗ -network at 0 iff X is countable. As applications we obtain (1) C p ( X ) has the strong Pytkeev property introduced by Tsaban and Zdomskyy iff X is countable; (2) C p ( X ) is an ℵ-space iff X is countable. Relating to the strong Pytkeev property, we study function spaces C p ( X ) and C k ( X ) with property ( # ) .