Infinite dimensional sequential compactness: Sequential compactness based on barriers
Canadian journal of mathematics, Tome 77 (2025) no. 6, pp. 2083-2110
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We introduce a generalization of sequential compactness using barriers on $\omega $ extending naturally the notion introduced in [W. Kubiś and P. Szeptycki, On a topological Ramsey theorem, Canad. Math. Bull., 66 (2023), 156–165]. We improve results from [C. Corral and O. Guzmán and C. López-Callejas, High dimensional sequential compactness, Fund. Math.] by building spaces that are ${\mathcal {B}}$-sequentially compact but not ${\mathcal {C}}$-sequentially compact when the barriers ${\mathcal {B}}$ and ${\mathcal {C}}$ satisfy certain rank assumption which turns out to be equivalent to a Katětov-order assumption. Such examples are constructed under the assumption ${\mathfrak {b}} ={\mathfrak {c}}$. We also exhibit some classes of spaces that are ${\mathcal {B}}$-sequentially compact for every barrier ${\mathcal {B}}$, including some classical classes of compact spaces from functional analysis, and as a byproduct, we obtain some results on angelic spaces. Finally, we introduce and compute some cardinal invariants naturally associated to barriers.
Mots-clés :
B-sequentially compact space, barrier, sequentially compact, Ramsey convergence, almost disjoint family, bounding number, Nash-Williams, bisequential
Corral, C.; Guzmán, O.; López-Callejas, C.; Memarpanahi, P.; Szeptycki, P.; Todorčević, S. Infinite dimensional sequential compactness: Sequential compactness based on barriers. Canadian journal of mathematics, Tome 77 (2025) no. 6, pp. 2083-2110. doi: 10.4153/S0008414X24000646
@article{10_4153_S0008414X24000646,
author = {Corral, C. and Guzm\'an, O. and L\'opez-Callejas, C. and Memarpanahi, P. and Szeptycki, P. and Todor\v{c}evi\'c, S.},
title = {Infinite dimensional sequential compactness: {Sequential} compactness based on barriers},
journal = {Canadian journal of mathematics},
pages = {2083--2110},
year = {2025},
volume = {77},
number = {6},
doi = {10.4153/S0008414X24000646},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X24000646/}
}
TY - JOUR AU - Corral, C. AU - Guzmán, O. AU - López-Callejas, C. AU - Memarpanahi, P. AU - Szeptycki, P. AU - Todorčević, S. TI - Infinite dimensional sequential compactness: Sequential compactness based on barriers JO - Canadian journal of mathematics PY - 2025 SP - 2083 EP - 2110 VL - 77 IS - 6 UR - http://geodesic.mathdoc.fr/articles/10.4153/S0008414X24000646/ DO - 10.4153/S0008414X24000646 ID - 10_4153_S0008414X24000646 ER -
%0 Journal Article %A Corral, C. %A Guzmán, O. %A López-Callejas, C. %A Memarpanahi, P. %A Szeptycki, P. %A Todorčević, S. %T Infinite dimensional sequential compactness: Sequential compactness based on barriers %J Canadian journal of mathematics %D 2025 %P 2083-2110 %V 77 %N 6 %U http://geodesic.mathdoc.fr/articles/10.4153/S0008414X24000646/ %R 10.4153/S0008414X24000646 %F 10_4153_S0008414X24000646
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