Domination by second countable spaces and Lindelof 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 subset of L implies A(K) subset of 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 Lindelof Sigma-space. Under MA(omega(1)), if X is compact and (X x X) \ Delta has a compact cover ordered by a Polish space then X is metrizable: here Delta = {(x, x): x is an element of X} is the diagonal of the space X. Besides, if X is a compact space of countable tightness and X-2\Delta belongs to M then X is metrizable in ZFC. We also 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 subset of X there exists F is an element of F with P subset of F. It is a ZFC result that if X is a compact space and (X x X) \ Delta 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. (C) 2010 Elsevier B.V. All rights reserved.