Infinite dimensional sequential compactness: Sequential compactness based on barriers

Abstract

We introduce a generalization of sequential compactness using barriers on ω 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 B-sequentially compact but not C-sequentially compact when the barriers B and C satisfy certain rank assumption which turns out to be equivalent to a Katětov-order assumption. Such examples are constructed under the assumption L = C . We also exhibit some classes of spaces that are B-sequentially compact for every barrier 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.