Borel hierarchy

Summary

The Borel hierarchy sorts the Borel sets of a Polish space into levels $\Sigma^0_\alpha$, $\Pi^0_\alpha$ and $\Delta^0_\alpha$, indexed by the countable ordinals $\alpha \ge 1$. It starts with the open and the closed sets, and each new level allows countable unions of complements of earlier sets. Every Borel set appears at some countable level, and in an uncountable Polish space every level contains new sets. The level of a set measures how complicated it is, and this is a basic tool of descriptive set theory.