---
tag: 4P4Z4ZGXCY
kind: concept
name: Borel hierarchy
msc: ["03E15", "54H05", "28A05"]
sources:
  - url: https://en.wikipedia.org/wiki/Borel_hierarchy
same_as:
  wikidata: Q893487
  wikipedia: Borel hierarchy
---

## Summary

The Borel hierarchy sorts the Borel sets of a [[Polish space|584GXXY73M]] into levels $\Sigma^0_\alpha$, $\Pi^0_\alpha$ and $\Delta^0_\alpha$, indexed by the countable [[ordinals|0YSCSX5YZC]] $\alpha \ge 1$.
It starts with the open and the closed sets, and each new level allows countable unions of complements of earlier sets.
Every [[Borel set|0WQXY32Q8D]] 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|TF8HMVXKYC]].

## Explanations

### L1 human

Open and closed sets are the simplest sets.
Countable unions of closed sets, and countable intersections of open sets, are one step more complex.
Repeating this, one climbs a ladder of complexity with one rung for each countable ordinal.
For example, the set of rational numbers is a countable union of points, so it sits on the second rung, but it is not a countable intersection of open sets.