Domination by second countable spaces and Lindelöf Σ-property

Given a space M, a family of sets A of a space X is ordered by M if A = { A K : K is a compact subset of M} and K ⊂ L implies A K ⊂ A L . We study the class M of spaces which have compact covers ordered by a second countable space. We prove that a space C p ( X ) belongs to M if and only if it is a Lindelöf Σ-space. Under MA ( ω 1 ) , if X is compact and ( X × X ) \ Δ has a compact cover ordered by a Polish space then X is metrizable; here Δ = { ( x , x ) : x ∈ X } is the diagonal of the space X. Besides, if X is a compact space of countable tightness and X 2 \ Δ belongs to M then X is metrizable in ZFC. o consider the class M ⁎ of spaces X which have a compact cover F ordered by a second countable space with the additional property that, for every compact set P ⊂ X there exists F ∈ F with P ⊂ F . It is a ZFC result that if X is a compact space and ( X × X ) \ Δ belongs to M ⁎ then X is metrizable. We also establish that, under CH, if X is compact and C p ( X ) belongs to M ⁎ then X is countable.