Property of Baire
Encyclopedia
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 Δ denotes the symmetric difference
).
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.
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.
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. 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.
of a topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
has the property of Baire (Baire property, named after René-Louis BaireRené-Louis Baire
René-Louis Baire was a French mathematician most famous for his Baire category theorem, which helped to generalize and prove future theorems...
), or is called an almost open set, if it differs from an open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
by a meager set; that is, if there is an open set
such that is meager (where Δ denotes the symmetric differenceSymmetric difference
In mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....
).
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
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
or intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of almost open sets is again almost open.
If a subset of a Polish space
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish...
has the property of Baire, then its corresponding Banach-Mazur game is determined
Determinacy
In set theory, a branch of mathematics, determinacy is the study of under what circumstances one or the other player of a game must have a winning strategy, and the consequences of the existence of such strategies.-Games:...
. The converse does not hold; however, if every game in a given adequate pointclass
Adequate pointclass
In the mathematical field of descriptive set theory, a pointclass can be called adequate if it contains all recursive pointsets and is closed under recursive substitution, bounded universal and existential quantification and preimages by recursive functions....
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
Large cardinal property
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large"...
, that every projective set (in a Polish space) has the property of Baire.
It follows from the axiom of choice that there are sets of reals without the property of Baire. In particular, the Vitali set
Vitali set
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by . The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence is proven on the assumption of the axiom of...
does not have the property of Baire. Already weaker versions of choice are sufficient: the Boolean prime ideal theorem
Boolean prime ideal theorem
In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given abstract algebra. A common example is the Boolean prime ideal theorem, which states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on...
implies that there is a nonprincipal ultrafilter
Ultrafilter
In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" or "almost nothing"...
on the set of natural numbers; each such ultrafilter induces, via binary representations of reals, a set of reals without the Baire property.