High dimensional countable compactness and ultrafilters

Abstract

We define several notions of a limit point on sequences with domain a barrier in $[ω]^{<ω}$ focusing on the two dimensional case $[ω]^2$. By exploring some natural candidates, we show that countable compactness has a number of generalizations in terms of limits of high dimensional sequences and define a particular notion of $α$-countable compactness for $α\leqω_1$. We then focus on dimension 2 and compare 2-countable compactness with notions previously studied in the literature. We present a number of counterexamples showing that these classes are different. In particular assuming the existence of a Ramsey ultrafilter, a subspace of $βω$ which is doubly countably compact whose square is not countably compact, answering a question of T. Banakh, S. Dimitrova and O. Gutik. The analysis of this construction leads to some possibly new types of ultrafilters related to discrete, P-points and Ramsey ultrafilters.

Figures (3)

• Figure 1: High dimensional countable compactness properties ($\alpha\in\omega_1$).
• Figure 2: Relations between 2 dimensional versions of countable compactness
• Figure 3: Special kinds of ultrafilters

Theorems & Definitions (44)

• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Theorem 2.6
• proof
• Corollary 2.7
• proof