Property of Baire

Difference of an open set by a meager set

A subset A {\displaystyle A} of a topological space X {\displaystyle X} 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 U ⊆ X {\displaystyle U\subseteq X} such that A △ U {\displaystyle A\bigtriangleup U} is meager (where △ {\displaystyle \bigtriangleup } denotes the symmetric difference).[1]

Definitions

A subset A ⊆ X {\displaystyle A\subseteq X} of a topological space X {\displaystyle X} is called almost open and is said to have the property of Baire or the Baire property if there is an open set U ⊆ X {\displaystyle U\subseteq X} such that A △ U {\displaystyle A\bigtriangleup U} is a meager subset, where △ {\displaystyle \bigtriangleup } denotes the symmetric difference.[1] Further, A {\displaystyle A} has the Baire property in the restricted sense if for every subset E {\displaystyle E} of X {\displaystyle X} the intersection A ∩ E {\displaystyle A\cap E} has the Baire property relative to E {\displaystyle E} .[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 Γ {\displaystyle \Gamma } is determined, then every set in Γ {\displaystyle \Gamma } 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. ^ a b c Oxtoby, John C. (1980), "4. The Property of Baire", Measure and Category, Graduate Texts in Mathematics, vol. 2 (2nd ed.), Springer-Verlag, pp. 19–21, ISBN 978-0-387-90508-2.
  2. ^ Kuratowski, Kazimierz (1966), Topology. Vol. 1, Academic Press and Polish Scientific Publishers.
  3. ^ Becker, Howard; Kechris, Alexander S. (1996), The descriptive set theory of Polish group actions, London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, Cambridge, p. 69, doi:10.1017/CBO9780511735264, ISBN 0-521-57605-9, MR 1425877.
  4. ^ Oxtoby (1980), p. 22.
  5. ^ Blass, Andreas (2010), "Ultrafilters and set theory", Ultrafilters across mathematics, Contemporary Mathematics, vol. 530, Providence, RI: American Mathematical Society, pp. 49–71, doi:10.1090/conm/530/10440, MR 2757533. See in particular p. 64.

External links

  • Springer Encyclopaedia of Mathematics article on Baire property