High dimensional countable compactness and ultrafilters
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| Format: | Preprint |
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2024
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| author | Corral, Cesar Memarpanahi, Pourya Szeptycki, Paul |
| author_facet | Corral, Cesar Memarpanahi, Pourya Szeptycki, Paul |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_17217 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | High dimensional countable compactness and ultrafilters Corral, Cesar Memarpanahi, Pourya Szeptycki, Paul General Topology Primary: 54D80, 54A20, 54D35, 03E35 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. |
| title | High dimensional countable compactness and ultrafilters |
| topic | General Topology Primary: 54D80, 54A20, 54D35, 03E35 |
| url | https://arxiv.org/abs/2406.17217 |