Some characterizations for υX to be Lindelöf Σ or K-analytic in terms of

If X is a completely regular space it is proved that (i) υX is Lindelöf Σ if and only if there exists a covering { A α : α ∈ Σ } of C ( X ) indexed by a subset Σ of N N such that if A ( α | n ) = ⋃ { A β : β ∈ Σ , β ( i ) = α ( i ) , 1 ⩽ i ⩽ n } for each ( α , n ) ∈ Σ × N and U is a balanced neighborhood of the origin in C p ( X ) then for each α ∈ N N there is n ∈ N such that A ( α | n ) ⊆ n U , 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 { ϵ V α : α ∈ N N , ϵ > 0 } with V β ⊆ V α if α ⩽ β . We also show that (iii) if C p ( 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.