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

In this note we firstly show that if X is a σ-product of Lindelöf Σ-spaces, then C p ( X ) is a semi-monotonically monolithic space and hence it is monolithic. second part of this note we show that if { X α : α ∈ Λ } is a family of topological spaces such that ∏ i ⩽ n X α i is a D-space for each n ∈ N and α i ∈ Λ for each i ⩽ n , then any σ-product of { X α : 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 C p ( Y ) is monotonically monolithic, then C p ( X ) is monotonically monolithic. 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.