Cp(X) and Arhangelʹskiǐʹs αi-spaces

Nogura showed that whereas Arhangelʹskiǐʹs properties α1, α2 and α3 are preserved by finite products, the property α4 is not. It is shown here that for each space X the properties α2, α3 and α4 are the same for the function space Cp(X). As a consequence, α4 is closed under finite products of such function spaces.