# Property of Baire

A subset of a topological space has the **property of Baire** (**Baire property**, named after René-Louis Baire), or is called an **almost open** set, if it differs from an open set by a meager set; that is, if there is an open set such that is meager (where denotes the symmetric difference).[1] Further, has the **Baire property in the restricted sense** if for every subset of the intersection has the Baire property relative to . [2]

The family of sets with the property of Baire forms a σ-algebra. That is, the complement of an almost open set is almost open, and any countable union or intersection of almost open sets is again almost open.[1] Since every open set is almost open (the empty set is meager), it follows that every Borel set is almost open.

If a subset of a Polish space has the property of Baire, then its corresponding Banach–Mazur game is determined. The converse does not hold; however, if every game in a given adequate pointclass **Γ** is determined, then every set in **Γ** has the property of Baire. Therefore, it follows from projective determinacy, which in turn follows from sufficient large cardinals, that every projective set (in a Polish space) has the property of Baire.[3]

It follows from the axiom of choice that there are sets of reals without the property of Baire. In particular, the Vitali set does not have the property of Baire.[4] Already weaker versions of choice are sufficient: the Boolean prime ideal theorem implies that there is a nonprincipal ultrafilter on the set of natural numbers; each such ultrafilter induces, via binary representations of reals, a set of reals without the Baire property.[5]

## See also

## References

- Oxtoby, John C. (1980), "4. The Property of Baire",
*Measure and Category*, Graduate Texts in Mathematics,**2**(2nd ed.), Springer-Verlag, pp. 19–21, ISBN 978-0-387-90508-2. - Kuratowski, Kazimierz (1966),
*Topology. Vol. 1*, Academic Press and Polish Scientific Publishers. - Becker, Howard; Kechris, Alexander S. (1996),
*The descriptive set theory of Polish group actions*, London Mathematical Society Lecture Note Series,**232**, Cambridge University Press, Cambridge, p. 69, doi:10.1017/CBO9780511735264, ISBN 0-521-57605-9, MR 1425877. - Oxtoby (1980), p. 22.
- Blass, Andreas (2010), "Ultrafilters and set theory",
*Ultrafilters across mathematics*, Contemporary Mathematics,**530**, Providence, RI: American Mathematical Society, pp. 49–71, doi:10.1090/conm/530/10440, MR 2757533. See in particular p. 64.