Property of Baire
A subset [math]\displaystyle{ A }[/math] of a topological space [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ U\subseteq X }[/math] such that [math]\displaystyle{ A \bigtriangleup U }[/math] is meager (where [math]\displaystyle{ \bigtriangleup }[/math] denotes the symmetric difference).[1]
Definitions
A subset [math]\displaystyle{ A \subseteq X }[/math] of a topological space [math]\displaystyle{ X }[/math] is called almost open and is said to have the property of Baire or the Baire property if there is an open set [math]\displaystyle{ U\subseteq X }[/math] such that [math]\displaystyle{ A \bigtriangleup U }[/math] is a meager subset, where [math]\displaystyle{ \bigtriangleup }[/math] denotes the symmetric difference.[1] Further, [math]\displaystyle{ A }[/math] has the Baire property in the restricted sense if for every subset [math]\displaystyle{ E }[/math] of [math]\displaystyle{ X }[/math] the intersection [math]\displaystyle{ A\cap E }[/math] has the Baire property relative to [math]\displaystyle{ E }[/math].[2]
Properties
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 [math]\displaystyle{ \Gamma }[/math] is determined, then every set in [math]\displaystyle{ \Gamma }[/math] 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
- Almost open map – Map that satisfies a condition similar to that of being an open map.
- Baire category theorem – On topological spaces where the intersection of countably many dense open sets is dense
- Open set – Basic subset of a topological space
References
- ↑ 1.0 1.1 1.2 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, https://books.google.com/books?id=wUDjoT5xIFAC&pg=PA19.
- ↑ 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, https://books.google.com/books?id=L4Jf_ZRxqt8C&pg=PA69.
- ↑ (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. See in particular p. 64.
External links
 |  |
