On some monotone properties (Q2254718)

In this article some results obtained in [\textit{M. Cúth}, Topology Appl. 171, 87--90 (2014; Zbl 1323.54019), \textit{M. Cúth} and \textit{O. F. K. Kalenda}, J. Math. Anal. Appl. 423, No. 1, 18--31 (2015; Zbl 1312.54005), \textit{G. Gruenhage}, Topology Appl. 17, 287--304 (1984; Zbl 0547.54016), \textit{R. Rojas-Hernández} and \textit{Á. Tamariz-Mascarúa}, ibid. 159, No. 16, 3379--3391 (2012; Zbl 1258.54004)], are generalized. It is shown that any closed subspace in the \(\Sigma \)-product of cosmic spaces is monotonically stable. A space \(X\) is monotonically retractable if and only if it is monotonically \(\omega \)-stable and has a full retractional skeleton. It is proved that any monotonically retractable and monotonically \(\omega \)-monolithic space is monotonically Sokolov, and as a consequence, any monotonically Sokolov and monotonically \(\omega \)-stable space is monotonically retractable. Any closed subspace of a countable compact monotonically retractable space \(X\) is a \(W\)-set in \(X\).