A monotone version of the Sokolov property and monotone retractability in function spaces

We introduce the monotone Sokolov property and show that it is dual to monotone retractability in the sense that X   is monotonically retractable if and only if Cp(X)Cp(X) is monotonically Sokolov. Besides, a space X   is monotonically Sokolov if and only if Cp(X)Cp(X) is monotonically retractable. Monotone retractability and monotone Sokolov property are shown to be preserved by RR-quotient images and FσFσ-subspaces. Furthermore, every monotonically retractable space is Sokolov so it is collectionwise normal and has countable extent. We also establish that if X   and Cp(X)Cp(X) are Lindelöf Σ-spaces then they are both monotonically retractable and have the monotone Sokolov property. An example is given of a space X   such that Cp(X)Cp(X) has the Lindelöf Σ-property but neither X   nor Cp(X)Cp(X) is monotonically retractable. We also establish that every Lindelöf Σ-space with a unique non-isolated point is monotonically retractable. On the other hand, each Lindelöf space with a unique non-isolated point is monotonically Sokolov.