---
tag: TF8HMVXKYC
kind: concept
name: Descriptive set theory
msc: ["03E15", "54H05"]
sources:
  - url: https://en.wikipedia.org/wiki/Descriptive_set_theory
same_as:
  wikidata: Q225869
  wikipedia: Descriptive set theory
---

## Summary

Descriptive set theory studies the definable subsets of [[Polish spaces|584GXXY73M]] such as the real line, the Cantor space and the [[Baire space|P2WS16HKD7]].
It sorts sets by how complicated their definitions are, in the [[Borel hierarchy|4P4Z4ZGXCY]] and the [[projective hierarchy|8DZTNVTGY2]].
Its classical results show that simply defined sets, such as Borel and [[analytic sets|3RWWZYPAFV]], behave well.
For example, they are Lebesgue measurable, and they are either countable or contain a perfect set.
For more complex sets these questions depend on axioms beyond ZFC, such as [[determinacy|QXECDPAYTN]] and [[large cardinals|X1FG7PSCTY]].

## Explanations

### L1 human

Some sets of real numbers are easy to describe, like intervals or the set of rational numbers.
Others are only given by long chains of operations, or not by any explicit rule at all.
Descriptive set theory measures how hard a set is to describe, and shows that easy sets behave well.
For example, an uncountable closed set of reals always contains a copy of the Cantor set, so it has the size of the continuum.