Normal covers of various products

Throughout this paper, we consider the following two problems: (A) When does a rectangular normal cover of a product X×Y (or an infinite product ∏λ∈ΛXλ) have a σ-locally finite rectangular cozero refinement? (B) What kind of a refinement makes a rectangular open cover of a product X×Y (or an infinite product ∏λ∈ΛXλ) be normal? We shall discuss these problems on various products listed below.