Forbidden rectangles in compacta

Abstract

We establish negative results about “rectangular” local bases in compacta. For example, there is no compactum where all points have local bases of cofinal type ω×ω2. For another, the compactum βω has no nontrivially rectangular local bases, and the same is consistently true of βω∖ω: no local base in βω has cofinal type κ×c if κ<mσ-n-linked for some n∈[1,ω). Also, CH implies that every local base in βω∖ω has the same cofinal type as one in βω.We also answer a question of Dobrinen and Todorčević about cofinal types of ultrafilters: the Fubini square of a filter on ω always has the same cofinal type as its Fubini cube. Moreover, the Fubini product of nonprincipal P-filters on ω is commutative modulo cofinal equivalence.