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We prove that if X is a strongly zero-dimensional space, then for every locally compact second-countable space M, Cp(X, M) is a continuous image of a closed subspace of Cp(X). It follows in particular, that for strongly zero-dimensional spaces X, the Lindelöf number of Cp(X)×Cp(X) coincides with the Lindelöf number of Cp(X). We also prove that l(Cp(Xn)κ) ≤ l(Cp(X)κ) whenever κ is an infinite cardinal and X is a strongly zero-dimensional union of at most κcompact subspaces.

The Lindelöf number of C p(X)×C p(X) for strongly zero-dimensional X