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 [18,6,8,10].

Abstract. We give a new characterization of Valdivia compact spaces: A compact space is Valdivia if and only if it has a dense commutatively monotonically retractable subspace. This result solves Problem 5.12 from [6]. Besides, we introduce the notion of full c-skeleton and prove that a compact space is Corson if and only if it has a full c-skeleton.

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We show, in particular, that if nw(Nt) ≤ κ for any t ∈ T and C is a dense subspace of the product {Nt : t ∈ T } then, for any continuous (not necessarily surjective) map ϕ : C → K of C into a compact space K with t(K) ≤ κ, we have Ψ(ϕ(C)) ≤ κ. This result has several applications in Cp-theory. We prove, among other things, that if K is a non-metrizable Corson compact space then Cp(K) cannot be condensed onto a σ-compact space. This answers two questions published by Arhangel'skii and Pavlov.

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