Characterization of compact monotonically (\(\omega\)-)monolithic spaces using system of retractions (Q2452025)

Spaces with a rich family of retractions often occur both in topology and functional analysis. One of the possible concepts of a family of retrations was introduced in the paper [\textit{R. Rojas-Hernández} and \textit{V. V. Tkachuk}, J. Math. Anal. Appl. 412, No. 1, 125--137 (2014; Zbl 1308.54016)], in which spaces having such a system were called a monotonically Sokolov space. In this note it is shown that a compact space is monotonically Sokolov if and only if it is monotonically \(\omega\)-monolithic. This gives answers to several questions posed by R. Rojas-Hernández and V. V. Tkachuk.