Equalities for the extent of infinite products and Σ-products

For a space X, let e(X) = omega middot 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 = HA is an element of Lambda XA, when is e(X) = |Lambda| middot sup{e(XA) : lambda is an element of Lambda}? (2) For a Sigma-product Sigma of spaces XA, lambda is an element of Lambda, when is e(Sigma) = sup{e(XA) : lambda is an element of Lambda}? We show that the equalities in these questions hold if each XA is a strict p-space or a strong Sigma-space and, in the case of the first question, if the cardinality of the index set Lambda is less than the first weakly inaccessible. For semi-stratifiable spaces, we show that a slightly weaker form of these equalities holds. (C) 2021 Elsevier B.V. All rights reserved.