Infinite dimensional sequential compactness: Sequential compactness based on barriers

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Abstract 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.