Polish Spaces

A Polish space is a separable, completely metrizable topological space.

Borel Sets

A Borel set is a subset of a topological space that can be formed from open sets through countable unions, countable intersections, and complements.

Difficulty of Classification Problem For Torsion-Free Abelian Groups

The classification problem for the torsion-free abelian groups of rank $n\ge 2$ is geniunely harder than the classiciation problem for rank $n=1$, which can be partially answered using descriptive set theory and superrigidity theory.