Forbidden rectangles in compacta
arXiv:1208.3635 · doi:10.1016/j.topol.2012.06.004
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 ωx ω_2. For another, the compactum βωhas no nontrivially rectangular local bases, and the same is consistently true of βω\ ω: no local base in βωhas cofinal type κx c if κ< m_{σ-n-linked} for some n in [1,ω). Also, CH implies that every local base in βω\ ωhas the same cofinal type as one in βω. We also answer a question of Dobrinen and Todorcevic about cofinal types of ultrafilters: the Fubini square of a filter on ωalways has the same cofinal type as its Fubini cube. Moreover, the Fubini product of nonprincipal P-filters on ωis commutative modulo cofinal equivalence.
15 pages