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 Sigma(kappa)-product of a family of spaces with countable network weight, then C-p(X) is monotonically monolithic: (3) if X is the product of Lindelof Sigma-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 omega-monotonically monolithic space is monotonically monolithic. Moreover, we introduce the notion of monotonically <kappa-monolithic space and prove that any of these spaces X with nw(X) = kappa is hereditarily D. Finally, we prove that for every pseudocompact space X with network weight less or equal to omega(1), C-p(X) is hereditarily D. (C) 2012 Elsevier B.V. All rights reserved.