Some topological cardinal inequalities for spaces

Using the index of Nagami we get new topological cardinal inequalities for spaces C p ( X ) . A particular case of Theorem 1 states that if L ⊆ C p ( 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 ⊂ C p ( 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 .