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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].

In this article, we mainly study certain families of continuous retractions (r-skeletons) having certain rich properties. By using monotonically retractable spaces we solve a question posed by R. Z. Buzyakova in [6] concerning the Alexandroff duplicate of a space. Certainly, it is shown that if the space X has a full r-skeleton, then its Alexandroff duplicate also has a full r-skeleton and, in a very similar way, it is proved that the Alexandroff duplicate of a monotonically retractable space is monotonically retractable. The notion of q-skeleton is introduced and it is shown that every compact subspace of Cp(X) is Corson when X has a full q-skeleton. The notion of strong r-skeleton is also introduced to answer a question suggested by F. Casarrubias-Segura and R. Rojas-Hernández in their paper [7] by establishing that a space X is monotonically Sokolov iff it is monotonically ω-monolithic and has a strong r-skeleton. The techniques used here allow us to give a topological proof of a result of I. Bandlow [4] who used elementary submodels and uniform spaces.2010 Mathematics Subject Classification. Primary 54C99, 54C15, 54E20.

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