Domination by second countable spaces and Lindelöf Σ-property

Abstract: Given a space M, a family of sets of a space X is ordered by M if { is a compact subset of M} and implies . We study the class of spaces which have compact covers ordered by a second countable space. We prove that a space belongs to if and only if it is a Lindelöf Σ-space. Under , if X is compact and has a compact cover ordered by a Polish space then X is metrizable; here is the diagonal of the space X. Besides, if X is a compact space of countable tightness and belongs to then X is metrizable in ZFC. We also consider the class of spaces X which have a compact cover ordered by a second countable space with the additional property that, for every compact set there exists with . It is a ZFC result that if X is a compact space and belongs to then X is metrizable. We also establish that, under CH, if X is compact and belongs to then X is countable. [ABSTRACT FROM AUTHOR]