---
tag: FZ7237VY65
kind: concept
name: Baire property
msc: ["54E52", "03E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Property_of_Baire
same_as:
  wikidata: Q803931
  wikipedia: Property of Baire
---

## Summary

A subset of a topological space has the Baire property if it differs from an [[open set|02BH3MCBQB]] by a [[meagre set|0XGCZ4GDZ6]].
It is the topological analogue of Lebesgue measurability, with meagre sets in the role of null sets.
The sets with the Baire property form a sigma-algebra that contains all Borel sets, and in a Polish space it also contains all [[analytic sets|3RWWZYPAFV]].
Under the axiom of choice there are sets of reals without the Baire property.

## Explanations

### L1 human

Meagre sets are the "topologically small" sets, such as countable sets of reals or the Cantor set.
A set has the Baire property when, after ignoring a small set of errors, it looks like an open set.
For example, the closed interval $[0, 1]$ differs from the open interval $(0, 1)$ only by the two points $0$ and $1$, and a finite set is meagre.
So $[0, 1]$ has the Baire property.
Sets with this property behave well in arguments that use the [[Baire category theorem|0A7KGAYHGV]].