Products of sequentially compact spaces with no separability assumption

Let X be a product of topological spaces. We prove that X is sequentially compact if and only if all subproducts by ≤ s factors are sequentially compact. If s = h, then X is sequentially compact if and only if all factors are sequentially compact and all but at most < s factors are ultraconnected. We give a topological proof of the inequality cf s ≥ h. Recall that s denotes the splitting number and h the distributivity number. Some corresponding invariants are introduced, relative to an arbitrary topological property, more generally, relative to a subset of a partial infinitary semigroup.