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) C p ( C p ( X ) ) is monotonically monolithic if and only if X is monotonically monolithic; (2) if X is a Σ κ -product of a family of spaces with countable network weight, then C p ( X ) is monotonically monolithic; (3) if X is the product of Lindelöf Σ-spaces, then C p ( X ) is monotonically monolithic; (4) if C p ( X ) is monotonically monolithic and Y is a cozero subset of X, then C p ( Y ) is monotonically monolithic; (5) every compact ω-monotonically monolithic space is monotonically monolithic. er, 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 ω 1 , C p ( X ) is hereditarily D.