Equalities for the extent of infinite products and Σ-products

For a space X, let e(X)=ω⋅sup{|D|:D is a closed discrete subset in X}, which is called the extent of X. Here we deal with the following two questions: (1) For a product space X=∏_λ∈ΛX_λ, when is e(X)=|Λ|⋅sup{e(X_λ):λ∈Λ}? (2) For a Σ-product Σ of spaces X_λ,λ∈Λ, when is e(Σ)=sup{e(X_λ):λ∈Λ}? We show that the equalities in these questions hold if each X_λ is a strict p-space or a strong Σ-space and, in the case of the first question, if the cardinality of the index set Λ is less than the first weakly inaccessible. For semi-stratifiable spaces, we show that a slightly weaker form of these equalities holds.