Forbidden rectangles in compacta

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.

💡 Research Summary

The paper investigates the possible cofinal types of local bases in compact spaces, focusing on “rectangular” structures—local bases whose directed order is cofinal with a product order ω × κ. The authors first introduce the notion of a rectangular local base and explain why such bases are of interest in the study of compacta and ultrafilters.

The first main negative result shows that no compact space can have every point equipped with a local base of cofinal type ω × ω₂. The proof proceeds by assuming the existence of such a base at a point, extracting a decreasing sequence of closed neighborhoods, and demonstrating that the required ω₂‑length chain forces a violation of compactness (specifically, the finite subcover property). This extends earlier work where ω × ω bases are known to exist in certain spaces, establishing a sharp boundary at the first uncountable cardinal beyond ω₁.

The paper then turns to the Stone–Čech compactification βω of the natural numbers and its remainder βω \ ω. It is proved that βω admits no non‑trivial rectangular local bases: no point in βω has a local base cofinal with ω × κ for any κ < 𝔠 (the continuum). The argument uses the Tukey order of ultrafilters, showing that a rectangular base would imply the existence of a σ‑n‑linked ultrafilter of size below a certain cardinal invariant.

To formalize this obstruction, the authors introduce the cardinal invariant m_{σ‑n‑linked}, the minimal size of a family of subsets of ω that is σ‑n‑linked. They prove that if κ < m_{σ‑n‑linked} for some finite n ≥ 1, then βω \ ω cannot have a local base of type ω × κ. This result is consistent with ZFC but requires additional set‑theoretic assumptions for full verification.

Under the Continuum Hypothesis (CH), a stronger uniformity emerges: every local base in βω \ ω has the same cofinal type as some local base in βω. Consequently, CH collapses the spectrum of possible rectangular types in the remainder to those already realized in βω, eliminating any exotic ω × κ configurations.

The final section addresses questions about the cofinal types of ultrafilters under Fubini products. The authors prove two complementary theorems. First, for any filter ℱ on ω, the Fubini square ℱ ⊗ ℱ and the Fubini cube ℱ ⊗ ℱ ⊗ ℱ are cofinally equivalent; that is, their directed orders are Tukey‑equivalent. This shows that iterating the Fubini product does not increase the cofinal complexity of a filter. Second, when ℱ₁ and ℱ₂ are non‑principal P‑filters, their Fubini product is commutative modulo cofinal equivalence: ℱ₁ ⊗ ℱ₂ ≈ ℱ₂ ⊗ ℱ₁. The proof exploits the selective nature of P‑filters, which guarantees that any two directed systems can be interleaved without changing the Tukey type.

Overall, the paper establishes that certain “rectangular” cofinal types are forbidden in compacta, especially in βω and its remainder, and it clarifies how Fubini products interact with cofinal equivalence. These results tighten the connection between topological compactness, ultrafilter combinatorics, and cardinal invariants, and they answer a question of Dobrinen and Todorcevic concerning the Tukey types of Fubini powers of filters. The work opens further avenues for exploring which cofinal configurations can be realized under various set‑theoretic hypotheses.