Pathology of submeasures and $F_{\sigma }$ ideals

Abstract

We address some phenomena about the interaction between lower semicontinuous submeasures on $\mathbb{N}$ and $F_{\sigma }$ ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological $F_{\sigma }$ ideals. We give a partial answers to the question of whether every nonpathological tall $F_{\sigma }$ ideal is Katětov above the random ideal or at least has a Borel selector. Finally, we show a representation of nonpathological $F_{\sigma }$ ideals using sequences in Banach spaces.

arXiv admin note: substantial text overlap with arXiv:2111.10598