Some characterizations for υX to be Lindelöf Σ or K-analytic in terms of Cp(X)

If X is a completely regular space it is proved that (i) υX is Lindelöf Σ if and only if there exists a covering of C(X) indexed by a subset Σ of NN such that if for each (α,n)∈Σ×N and U is a balanced neighborhood of the origin in Cp(X) then for each α∈NN there is n∈N such that A(α|n)⊆nU, and (ii) υX is K-analytic if and only if there is in C(X) a locally convex topology stronger than the pointwise convergence topology with a base of absolutely convex neighborhoods of the origin of the form with Vβ⊆Vα if α⩽β. We also show that (iii) if Cp(X) is a Baire space with a closed Σ-covering of limited envelope, then X is countable. A number of applications of these results are given.