High dimensional sequential compactness

TL;DR

This paper constructs examples of high-dimensional sequentially compact spaces that lack lower-dimensional sequential compactness, improving previous results by using specific set-theoretic assumptions and introducing a new cardinal invariant.

Contribution

It provides new examples of n-sequentially compact spaces not (n+1)-sequentially compact, under various set-theoretic assumptions, and introduces a new splitting-like cardinal invariant.

Findings

Examples of n-sequentially compact spaces not (n+1)-sequentially compact

Improved constructions under $rak{b=c}$ and $rak{d}=rak{b}= ext{omega}_1$

Introduction of a new splitting-like cardinal invariant

Abstract

We give examples of $n$-sequentially compact spaces that are not $(n+1)$-sequentially compact under several assumptions. We improve results from Kubis and Szeptycki by building such examples from $\mathfrak{b=c}$ and $\diamondsuit(\mathfrak{b})+\mathfrak{d}=\omega_1$. We also introduce a new splitting-like cardinal invariant and then show that the same holds under $\mathfrak{s=b}$.