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HIGH DIMENSIONAL COUNTABLE COMPACTNESS AND ULTRAFILTERS
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 07 April 2025, pp. 1-21
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- Article
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We define several notions of a limit point on sequences with domain a barrier in
$[\omega ]^{<\omega }$ focusing on the two dimensional case 
$[\omega ]^2$. 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 
$\alpha $-countable compactness for 
$\alpha \leq \omega _1$. 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 
$\beta \omega $ which is doubly countably compact whose square is not countably compact, answering a question of T. Banakh, S. Dimitrova, and O. Gutik [3]. The analysis of this construction leads to some possibly new types of ultrafilters related to discrete, P-points and Ramsey ultrafilters.
