The product of two normal initially 𝜅-compact spaces

Abstract

We prove that it is independent from ZFC {\text {ZFC}} that for every cardinal κ \kappa the following statements are equivalent: (a) κ \kappa is singular; (b) initial κ \kappa -compactness (defined above the introduction) is productive; (c) initial κ \kappa -compactness is finitely productive; and (d) the product of two initially κ \kappa -compact normal spaces is initially κ \kappa -compact. In particular, MA \text {MA} implies that there are two countably compact normal spaces whose product is not countably compact.