Monotone pseudobase assignments and Lindelöf Σ-property

We introduce and study the spaces with κ-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 κ-monotone pseudo-network assignment is preserved by σ-products. It is also proved that a countably compact space X with an ω-monotone pseudobase assignment is compact and metrizable. If a countably compact space X has an ω-monotone pseudo-network assignment, then X is monotonically monolithic and hence Corson compact. In Lindelöf Σ-spaces, having a κ-monotone pseudo-network assignment is equivalent to being monotonically κ  -monolithic. As an application of the above results in CpCp-theory, we show that if CpCp(X)CpCp(X) is a Lindelöf Σ-space and s(X)=ωs(X)=ω, then X has a countable network; this solves an open problem published in 2001.