On monotone stability

The notion of monotonically monolithic space was introduced by V.V. Tkachuk in 2009 [8]. In this paper we introduce the notion of monotone stability and show that a space Cp(X)Cp(X) is monotonically monolithic if and only if X   is monotonically stable. As a consequence, a space Cp(X)Cp(X) is monotonically stable if and only if X   is monotonically monolithic. We also prove that Cp(X)Cp(X) is monotonically monolithic when X   is a ΣκΣκ-product of a family of Lindelöf Σ-spaces. These results answer some questions posed by Á. Tamariz-Mascarúa and the author in a previous paper [7].