The product of two normal initially 𝜅-compact spaces

Abstract

Abstract. We prove that it is independent from ZFC that for every cardinal k the following statements are equivalent:(a) K is singular; (b) initial K-compactness (defined above the introduction) is productive; (c) initial K-compactness is finitely productive; and (d) the product of two initially K-compact normal spaces is initially kcompact.In particular, MA implies that there are two countably compact normal spaces whose product is not countably compact.In our terminology Tx implies regular and normal, and k , X, and p denote infinite cardinals. Recall that a space is called initially K-compact if every open cover of cardinality at most k has a finite subcover, and that initial cocompactness is equivalent to countable compactness, for Tx -spaces. The meaning of initially < K-compact should be clear.