A note on D-property, monotone monolithicity and σ-product

Abstract

In this note we firstly show that if X is a σ-product of Lindelöf Σ-spaces, then Cp(X) is a semi-monotonically monolithic space and hence it is monolithic.In the second part of this note we show that if {Xα:α∈Λ} is a family of topological spaces such that ∏i⩽nXαi is a D-space for each n∈N and αi∈Λ for each i⩽n, then any σ-product of {Xα:α∈Λ} is a D-space. This generalizes a conclusion of A.V. Arhangelʼskii. By this conclusion and a known conclusion we have that any σ-product of regular weak θ¯-refinable (or (sub)metacompact) C-scattered spaces is a D-space. We also show that if Y is a dense subspace of a topological space X with |X∖Y|⩽ω such that Cp(Y) is monotonically monolithic, then Cp(X) is monotonically monolithic.In the last part of this note we show that if X has a k-in-countable weak base for some k∈N then X is a D-space.