The spaces Cp(X): decomposition into a countable union of bounded subspaces and completeness properties

AbstractA Tychonoff space X has to be finite if Cp(X) is σ-countably compact [23]. However, this is not true if only σ-pseudocompactness of Cp(X) is assumed. It is proved that Cp(X) is σ-pseudocompact iff X is pseudocompact and b-discrete. The technique developed yields an example showing that the theorem of Grothendieck [7] cannot be extended over the class of pseudocompact spaces. Some generalizations of the results of Lutzer and McCoy [9] are obtained. We establish also that ∏{Cp(Xt):tϵT} is a Baire space in case Cp(Xt) is Baire for each t ∈ T