Some topological cardinal inequalities for spaces Cp(X)Cp(X)

Using the index of Nagami we get new topological cardinal inequalities for spaces Cp(X)Cp(X). A particular case of Theorem 1 states that if L⊆Cp(X)L⊆Cp(X) is a Lindelöf Σ  -space and the Nagami index Nag(X)Nag(X) of X   is less or equal than the density d(L)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)Nag(Y)⩽Nag(X), L⊂Cp(Y)L⊂Cp(Y) and d(L)=d(Y)d(L)=d(Y); (ii) Y   admits a weaker completely regular Hausdorff topology τ′τ′ such that w(Y,τ′)⩽d(Y)=d(L)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 GG 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 GG.