A note on $G_{\delta}$ ideals of compact sets

TL;DR

This paper refines the representation of certain $G_{\delta}$ ideals of compact sets by showing the representing closed set can be chosen to be closed upwards, enhancing understanding of their structure.

Contribution

It demonstrates that the closed subset used in representing $G_{\delta}$ ideals can be assumed to be closed upwards, improving the existing representation framework.

Findings

Representation of $G_{\delta}$ ideals can be refined

Closed upwards sets can replace arbitrary closed sets in the representation

Enhances structural understanding of $G_{\delta}$ ideals

Abstract

Solecki has shown that a broad natural class of $G_{\delta}$ ideals of compact sets can be represented through the ideal of nowhere dense subsets of a closed subset of the hyperspace of compact sets. In this note we show that the closed subset in this representation can be taken to be closed upwards.