# 2011 seminar talk: Topologically invariant \sigma-ideals on Polish spaces

Talk held by Taras Banakh (Lviv State University, Ukraine) at the KGRC seminar on 2011-06-22.

### Abstract

We shall discuss the structure of topologically invariant
\(\sigma\)-ideals with Borel base on homogeneous Polish spaces.
An ideal \(\mathcal I\) of subsets of a topological space \(X\) is called *topologically invariant* if for each set \(A\in\mathcal I\) and
each homeomorphism \(h:X\to X\) we get \(h(A)\in\mathcal I\).

During the lecture I plan to cover the following topics:

(1) classification of topologically invariant \(\sigma\)-ideals on topologically homogeneous zero-dimensional Polish spaces;

(2) extremal (i.e., maximal, largest, smallest) topologically invariant \(\sigma\)-ideals on some "nice" Polish spaces;

(3) cardinal characteristics of topologically invariant \(\sigma\)-ideals on some "nice" Polish spaces and their interplay with the cardinal characteristics of the ideal \(\mathcal M\) of meager sets.