Monotone versions of some selection principles

Abstract A space X is monotonically Lindelof if there is an operator r that assigns to every open cover U a countable open cover r ( U ) so that r ( U ) refines U , and r ( U ) refines r ( V ) whenever U refines V . We consider the monotone versions of the multicover forms of the Lindelof property: Menger, Hurewicz, Rothberger.