Productivity numbers in topological spaces

The κ-productivity of classes C of topological spaces closed under quotients and disjoint sums is characterized by means of Cantor spaces. The smallest infinite cardinals κ such that such classes are not κ-productive are submeasurable cardinals. It follows that if a class of topological spaces is closed under quotients, disjoint sums and countable products, it is closed under products of non-sequentially many spaces (thus under all products, if sequential cardinals do not exist).