$G_{\delta}$ sets in $\sigma$-ideals generated by compact sets

TL;DR

This paper investigates how certain $G_{\delta}$ $\sigma$-ideals of compact sets in a Polish space can be represented through compact subsets of the hyperspace, even after extension by countable unions.

Contribution

It demonstrates that extended collections of $G_{\delta}$ or analytic sets covered by ideals can still be represented via compact subsets of the hyperspace.

Findings

Extended ideals remain representable via compact subsets of the hyperspace.

Representation persists after extending ideals by countable unions.

Provides a structural understanding of $G_{\delta}$ $\sigma$-ideals in Polish spaces.

Abstract

Given a compact Polish space $E$ and the hyperspace of its compact subsets $\mathcal{K}(E)$, we consider the class of $G_{\delta}$ $\sigma$-ideals of compact subsets of $E$ that can be represented via a compact subset of $\mathcal{K}(E)$. If we extend such an ideal $I$ by considering $G_{\delta}$ (or analytic) sets that are covered by countable unions of sets in $I$, we show that the extended collection can still be represented via some compact subset of $\mathcal{K}(E)$.