High dimensional sequential compactness

In this talk, we will explore a multidimensional version of sequential compactness introduced by Kubis and Szeptycki, known as n-sequential compactness (n-sc), where n is a natural number. They demonstrated that this property holds in compact metric spaces and showed that it induces a hierarchy of sequential compactness; that is, for any n, if a space X is (n+1)-sc, then it is also n-sc. The question they pose is whether this hierarchy is strict—specifically, whether for each n, it is possible to construct a space that is n-sc but not (n+1)-sc. In this presentation, we will discuss some recent progress on this question and mention further generalizations of sequential compactness to any countable ordinal.