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 C p ( X ) is monotonically Sokolov. Besides, a space X is monotonically Sokolov if and only if C p ( X ) is monotonically retractable. Monotone retractability and monotone Sokolov property are shown to be preserved by R -quotient images and F σ -subspaces. Furthermore, every monotonically retractable space is Sokolov so it is collectionwise normal and has countable extent. We also establish that if X and C p ( 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 C p ( X ) has the Lindelöf Σ-property but neither X nor C p ( 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.