D-property, monotone monolithicity and function spaces

In this article we answer some questions related to monotonically monolithic spaces posed in O.T. Alas et al. (2009) [1], V.V. Tkachuk (2009) [17], and V.V. Tkachuk (2012) [18]. We prove: (1) Cp(Cp(X)) is monotonically monolithic if and only if X is monotonically monolithic; (2) if X is a Σκ-product of a family of spaces with countable network weight, then Cp(X) is monotonically monolithic; (3) if X is the product of Lindelöf Σ-spaces, then Cp(X) is monotonically monolithic; (4) if Cp(X) is monotonically monolithic and Y is a cozero subset of X, then Cp(Y) is monotonically monolithic; (5) every compact ω-monotonically monolithic space is monotonically monolithic.Moreover, we introduce the notion of monotonically <κ-monolithic space and prove that any of these spaces X with nw(X)=κ is hereditarily D. Finally, we prove that for every pseudocompact space X with network weight less or equal to ω1, Cp(X) is hereditarily D.