Cent. Eur. J. Math. • 9(5) • 2011 • 978-983 DOI: 10.2478/s11533-011-0050-y The Lindelöf number of C p (X ) ×C p (X ) for strongly zero-dimensional X Oleg Okunev 1∗ 1 Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio y Rio Verde s/n, colonia San Manuel, Ciudad Universitaria, CP 72570 Puebla, Puebla, México We prove that if X is a strongly zero-dimensional space, then for every locally compact second-countable space M, C p (X,M) is a continuous image of a closed subspace of C p (X ). It follows in particular, that for strongly zero- dimensional spaces X , the Lindelöf number of C p (X ) ×C p (X ) coincides with the Lindelöf number of C p (X ). We also prove that l (C p (X n ) κ ) ≤ l (C p (X ) κ ) whenever κ is an infinite cardinal and X is a strongly zero-dimensional union of at most κ compact subspaces. 54C35 Topology of pointwise convergence • Lindelöf number © Versita Sp. z o.o. All spaces considered in this article are assumed to be Tychonoff (= completely regular Hausdorff). We use terminology and notation as in [3], except that the Lindelöf number of a space X is denoted by l(X ). The symbol ω denotes the set of non-negative integers, always considered with the discrete topology. We study the spaces C p (X,Z ) of all continuous functions on a space X with values in a space Z , equipped with the topology of pointwise convergence (see [2] for a thorough presentation of the theory of spaces of functions equipped with this topology). The space C p (X, R) is denoted by C p (X ). We often use the fact that if Y is a closed subspace of Z , then C p (X,Y ) is naturally homeomorphic to a closed subspace of C p (X,Z ). Recall that a space X is called zero- dimensional if X has a base consisting of clopen sets, and X is strongly zero-dimensional if every finite functionally open cover of X has a finite open disjoint refinement (that is, the Lebesgue covering dimension dim of the space X is equal to 0). It is well-known that X is strongly zero-dimensional if and only if the Stone– ˇ Cech compactification βX of X is zero-dimensional, and that every Lindelöf zero-dimensional space is strongly zero-dimensional [3, Sections 6.3 and 7.1]. ∗ E-mail: olegome@gmail.com