Variations on tightness in function spaces

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. It is known that some topological properties of Cp(X) can be well characterized by topological properties of X. For example, Arhangelʹskii and Pytkeev proved that the tightness of Cp(X) is countable if and only if every finite power of X is Lindelِf. In this paper, we give topological properties of X which characterize T-tightness and set-tightness of Cp(X). A characterization of set-tightness of Cp(X) answers a question posed by Bella.