Monotone pseudobase assignments and Lindelof Σ-property

We introduce and study the spaces with kappa-monotone pseudo-network (pseudobase) assignment. We show that the respective classes are invariant under arbitrary subspaces, countable products, and are lifted by condensations. Besides, the class of spaces with a kappa-monotone pseudo-network assignment is preserved by sigma-products. It is also proved that a countably compact space X with an omega-monotone pseudobase assignment is compact and metrizable. If a countably compact space X has an omega-monotone pseudo-network assignment, then X is monotonically monolithic and hence Corson compact. In Lindelof Sigma-spaces, having a kappa-monotone pseudo-network assignment is equivalent to being monotonically kappa-monolithic. As an application of the above results in C-p-theory, we show that if CpCp(X) is a Lindelof Sigma-space and s(X) = omega, then X has a countable network; this solves an open problem published in 2001. (C) 2016 Elsevier B.V. All rights reserved.