Baire property

2020 Mathematics Subject Classification: Primary: 54E52 [MSN][ZBL]

of a subset of a topological space

A property analogous to the measurability of a set. A subset $A$ of a topological space $X$ is said to have the Baire property if there is an open set $U$ such that the symmetric difference $(U\setminus A)\cup (A\setminus U)$ is a set of first category, i.e. it is a countable union of nowhere dense sets. Cp. with Chapter 4 of [Ox]. Some authors use the terminology almost open.

The following are useful characterizations.

Theorem 1 A set $A$ has the Baire property if and only if there is a closed set $C$ such that $(C\setminus A)\cup (A\setminus C)$ is of first category, cp. with Theorem 4.1 of [Ox].

Theorem 2 A set $A$ has the Baire property if and only if it is the union of a $G_\delta$ and a set of first category (resp. if and only if it there is a set of first category $B$ such that $A\cup B$ is an $F_{\sigma}$). Cp. with Theorem 4.4 of [Ox].

It follows easily from the definition that any countable union of sets with the Baire property has the Baire property. Moreover, it follows from Theorem 1 that the complement of a set with the Baire property has the Baire property. Therefore, the family of subsets of a topological space $X$ with the Baire property is a $\sigma$-algebra.

Sets with the Baire property can also be represented as differences of regular open sets and sets of first category (an open set is called regular if it coincides with the interior of its closure).

Moreover, on subsets of $\mathbb R$ with the Baire property the Banach-Mazur game is determined (cp. with Theorem 6.3 of [Ox]).

The axiom of choice implies the existence of subsets of the real numbers without the Baire property: the Vitali set is a classical example.

References

[Ba]	R. Baire, "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars (1905) Zbl 36.0438.01
[Ba]	E. Čech, "Topological spaces" , Wiley (1966) MR0211373 Zbl 0141.39401
[Ku]	K. Kuratowski, "Topology" , 1–2 , Acad. Press (1966–1968) MR0259835 MR0217751 Zbl 0158.40802
[Ox]	J.C. Oxtoby, "Measure and category" , Springer (1971) MR0393403 0217.09201 Zbl 0217.09201
[Ro]	H.L. Royden, "Real analysis", Macmillan (1968) MR0151555 Zbl 0197.03501