Metrizable compacta in the space of continuous functions with the topology of pointwise convergence

We prove that every point-finite family of nonempty functionally open sets in a topological space X has the cardinality at most an infinite cardinal κ if and only if w(X) ≦ κ for every Valdvia compact space Y C _p (X). Correspondingly a Valdivia compact space Y has the weight at most an infinite cardinal κ if and only if every point-finite family of nonempty open sets in C _p (Y) has the cardinality at most κ, that is p(C _p (Y)) ≦ κ. Besides, it was proved that w(Y) = p(C _p (Y)) for every linearly ordered compact Y. In particular, a Valdivia compact space or linearly ordered compact space Y is metrizable if and only if p(C _p (Y)) = ℵ₀. This gives answer to a question of O. Okunev and V. Tkachuk.