---
tag: FDW81MBGJK
kind: concept
name: Infinity-Borel set
aliases: [infinity-borel, borel]
sources:
  - wikipedia: en:2234812@r1191808170
    locator: Infinity-Borel set
    note: adopted
    mention: wikipedia://en/2234812
    sections: [Summary]
same_as:
  wikidata: Q17145837
  wikipedia: Infinity-Borel set
inventory: [D102XNYWEY]
---

## Summary

In set theory, a subset of a Polish space $X$ is **∞-Borel** if it can be obtained by starting with the [[open subsets|01ZQQDCERP]] of $X$, and transfinitely iterating the operations of [[complementation|0RKF1ZT5JJ]] and well-ordered [[union|0X1GSWNBN1]]. This concept is usually considered without the assumption of the [[axiom of choice|0HTGB13FE2]], which means that the ∞-Borel sets may fail to be closed under well-ordered union; see below.