Function spaces with a countable cs∗-network at a point

For a Tychonoff space X, we denote by Cp(X) (Ck(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 Cp(X) has a countable cs∗-network at 0 iff X is countable. As applications we obtain (1) Cp(X) has the strong Pytkeev property introduced by Tsaban and Zdomskyy iff X is countable; (2) Cp(X) is an ℵ-space iff X is countable. Relating to the strong Pytkeev property, we study function spaces Cp(X) and Ck(X) with property (#).