Some topological cardinal inequalities for spaces Cp(X)

Using the index of Nagami we get new topological cardinal inequalities for spaces Cp(X). A particular case of Theorem 1 states that if L ⊆ Cp(X) is a Lindelöf Σ-space and the Nagami index Nag(X) of X is less or equal than the density d(L) of L (which holds for instance if X is a Lindelöf Σ-space), then (i) there exists a completely regular Hausdorff space Y such that Nag(Y ) Nag(X), L ⊂ Cp(Y ) and d(L) = d(Y ); (ii) Y admits a weaker completely regular Hausdorff topology τ such that w(Y , τ
) d(Y ) = d(L). This applies, among other things, to characterize analytic sets for the weak topology of any locally convex space E in a large class G of locally convex spaces that includes (DF)-spaces and (LF)-spaces. The latter yields a result of Cascales–Orihuela about weak metrizability of weakly compact sets in spaces from the class G.