Property of Baire

A subset $A$ of a topological space $X$ 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 $U\subseteq X$ such that $A\bigtriangleup U$ is meager (where $\bigtriangleup$ denotes the symmetric difference).^[1] Further, $A$ has the Baire property in the restricted sense if for every subset $E$ of $X$ the intersection $A\cap E$ has the Baire property relative to $E$ . ^[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]

References

^ ^a ^b 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.

External links

Springer Encyclopaedia of Mathematics article on Baire property

[oxtoby-1] 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.

[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, 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, 530, Providence, RI: American Mathematical Society, pp. 49–71, doi:10.1090/conm/530/10440, MR 2757533. See in particular p. 64.

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Property of Baire

See also

References

External links