On some monotone properties

In this paper we deal with some classes of spaces defined by networks and retractions, in particular we prove: Any closed subspace in a Σ-product of cosmic spaces is monotonically stable. A space X is monotonically retractable if and only if it is monotonically ω-stable and has a full retractional skeleton. Any monotonically retractable and monotonically ω-monolithic space is monotonically Sokolov, and as a consequence, any monotonically Sokolov and monotonically ω-stable space is monotonically retractable. Any closed subspace of a countably compact monotonically retractable space X is a W-set in X. These results generalize some results obtained in [18], [6], [8] and [10].