Families of continuous retractions and function spaces

In this paper, we mainly study certain families of continuous retractions (r-skeletons) having some rich properties. It is shown that if the space X has a full r-skeleton, then its Alexandroff duplicate also has a full r-skeleton. In a very similar way, it is proved that the Alexandroff duplicate of a monotonically retractable space is monotonically retractable. By using monotonically retractable spaces, we solve a question posed by R.Z. Buzyakova in [7] concerning the Alexandroff duplicate of a space. The notion of q  -skeleton is introduced and it is shown that every compact subspace of Cp(X)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 the second author in [8]. Certainly, it is established that a space X is monotonically Sokolov if and only if 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 [5] who used elementary submodels and uniform spaces.