A special point from ⋄ and strongly ω-bounded spaces

A Tychonoff space is CNP if it is a P-set in its Stone-Čech compactification. We are interested in the question of whether the property of being a CNP space is finitely productive. A space X is strongly ω-bounded if every σ-compact subset of X has compact closure in X. In this paper, we show that the existence of two CNP spaces whose product is not CNP is equivalent to the existence of a space which is not strongly ω-bounded but which is the union of two subsets each of which is strongly ω-bounded. We use ⋄ to construct a special point in β(ω×(βω∖ω)) and use that point to find a non-strongly ω-bounded space which is a union of two strongly ω-bounded subsets.