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-sigma-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 Lindelof Sigma-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 Lindelof Sigma-property but neither X nor C-p(X) is monotonically retractable. We also establish that every Lindelof Sigma-space with a unique non-isolated point is monotonically retractable. On the other hand, each Lindelof space with a unique non-isolated point is monotonically Sokolov. (C) 2013 Elsevier Inc. All rights reserved.