# On some relations between properties of invariant $\sigma$-ideals in Polish spaces

## TL;DR

This paper explores properties of invariant -ideals in Polish spaces, establishing their relationships and characterizations, including conditions for nonmeasurability and the Fubini Property.

## Contribution

It proves that -cc -ideals are tall, links the Smital Property to non(), and characterizes nonmeasurable -Luzin sets, advancing understanding of ideal properties.

## Key findings

- -cc -ideals are tall
- Weaker Smital Property implies non() containment
- The ideal of countable subsets of  does not have the Fubini Property

## Abstract

In this paper we shall consider a couple of properties of $\sigma$-ideals and study relations between them. Namely we will prove that $\mathfrak{c}$-cc $\sigma$-ideals are tall and that the Weaker Smital Property implies that every Borel $\mathcal{I}$-positive set contains a witness for non($\mathcal{I}$) as well, as satisfying ccc and Fubini Property. We give also a characterization of nonmeasurability of $\mathcal{I}$-Luzin sets and prove that the ideal $[\mathbb{R}]^{\leq\omega}$ does not posses the Fubini Property using some interesting lemma about perfect sets.