Hereditarily monotonically Sokolov spaces have countable network weight

摘要

We provide an example to show that the monotone Sokolov property is not necessarily preserved under compact continuous images. Furthermore, we prove that if X-2 Delta(X) 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 Lindelof Sigma-property in C-p-spaces on Alexandrov doubles, and show that the L Sigma(L Sigma( = omega ))-property is equivalent to the L Sigma( = omega )-property for the class of C-p-spaces and for the class of Gul'ko spaces. These results solve some problems published by Kalenda [7], Tkachuk [23], Garcia-Ferreira and Rojas-Hernandez [5], and Molina-Lara and Okunev [11]. (C) 2019 Elsevier B.V. All rights reserved.