Topologically invariant $\sigma$-ideals on the Hilbert cube

TL;DR

This paper classifies topologically invariant $\sigma$-ideals with Borel bases on the Hilbert cube, compares their cardinal characteristics, and resolves a problem about minimal covering families of Cantor sets.

Contribution

It provides a classification of such ideals and establishes that the minimal cardinalities of Cantor set coverings are equal for the unit interval and the Hilbert cube.

Findings

Classified topologically invariant $\sigma$-ideals with Borel bases on the Hilbert cube.

Proved the minimal cardinalities of Cantor set coverings are the same for the unit interval and the Hilbert cube.

Abstract

We study and classify topologically invariant $\sigma$-ideals with a Borel base on the Hilbert cube and evaluate their cardinal characteristics. One of the results of this paper solves (positively) a known problem whether the minimal cardinalities of the families of Cantor sets covering the unit interval and the Hilbert cube are the same.