On the Pytkeev property in spaces of continuous functions (II)

We prove that for each Polish space X, the space C(X) of continuous real-valued functions on X satisfies a strong version of the Pytkeev property, if endowed with the compact-open topology. (This shows that whereas it need not be metrizable, it is "very close" to that.) We also consider the Pytkeev property in the case where C(X) is endowed with the topology of pointwise convergence.