Hereditarily monotonically Sokolov spaces have countable network weight

Abstract

We provide an example to show that the monotone Sokolov property is not necessarily preserved under compact continuous images. Furthermore, we prove that if X2∖ΔX is either monotonically Sokolov or monotonically retractable, then X must be cosmic; and that if X is either hereditarily monotonically Sokolov or hereditarily monotonically retractable, then X has a countable network. Moreover, we characterize the Lindelöf Σ-property in Cp-spaces on Alexandrov doubles, and show that the LΣ(LΣ(⩽ω))-property is equivalent to the LΣ(⩽ω)-property for the class of Cp-spaces and for the class of Gul'ko spaces. These results solve some problems published by Kalenda [7], Tkachuk [23], García-Ferreira and Rojas-Hernández [5], and Molina-Lara and Okunev [11].