Forbidden rectangles in compacta

TL;DR

This paper investigates the structure of local bases in compact spaces, demonstrating limitations on their cofinal types and exploring properties of ultrafilters and Fubini products, with implications for topology and set theory.

Contribution

It establishes new negative results about rectangular local bases in compacta and answers open questions regarding ultrafilter cofinal types and Fubini products.

Findings

No compactum has all points with local bases of cofinal type ω x ω₂.

βω has no nontrivially rectangular local bases.

Fubini square and cube of a filter share the same cofinal type.

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 \omega x \omega_2. For another, the compactum \beta\omega has no nontrivially rectangular local bases, and the same is consistently true of \beta\omega \ \omega: no local base in \beta\omega has cofinal type \kappa x c if \kappa < m_{\sigma-n-linked} for some n in [1,\omega). Also, CH implies that every local base in \beta\omega \ \omega has the same cofinal type as one in \beta\omega. We also answer a question of Dobrinen and Todorcevic about cofinal types of ultrafilters: the Fubini square of a filter on \omega always has the same cofinal type as its Fubini cube. Moreover, the Fubini product of nonprincipal P-filters on \omega is commutative modulo cofinal equivalence.