A
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
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
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
by a
meager set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
; that is, if there is an open set
such that
is meager (where
denotes the
symmetric difference
In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \.
Th ...
).
[.]
Definitions
A subset
of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
is called almost open and is said to have the property of Baire or the Baire property if there is an open set
such that
is a
meager subset, where
denotes the
symmetric difference
In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \.
Th ...
.
Further,
has the Baire property in the restricted sense if for every subset
of
the intersection
has the Baire property relative to
.
Properties
The family of sets with the property of Baire forms a
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
. That is, the
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
of an almost open set is almost open, and any
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
union or
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
of almost open sets is again almost open.
Since every open set is almost open (the empty set is meager), it follows that every
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...
is almost open.
If a subset of a
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 be ...
has the property of Baire, then its corresponding
Banach–Mazur game
In general topology, set theory and game theory, a Banach– Mazur game is a topological game played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept of Baire spa ...
is
determined. The converse does not hold; however, if every game in a given
adequate pointclass In the mathematical field of descriptive set theory, a pointclass
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a ''point'' is ordinarily understood to be an element of some perfect ...
is determined, then every set in
has the property of Baire. Therefore, it follows from
projective determinacy In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only to projective sets.
The axiom of projective determinacy, abbreviated PD, states that for any two-player infinite game of perfect informatio ...
, which in turn follows from sufficient
large cardinals
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" (for example, bigger than the least ...
, that every
projective set
In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is \boldsymbol^1_n for some positive integer n. Here A is
* \boldsymbol^1_1 if A is analytic
* \boldsymbol^1_n if the complement of A, X\se ...
(in a Polish space) has the property of Baire.
It follows from the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
that there are sets of
reals without the property of Baire. In particular, the
Vitali set
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vita ...
does not have the property of Baire. Already weaker versions of choice are sufficient: the
Boolean prime ideal theorem
In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by cons ...
implies that there is a nonprincipal
ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...
on the set of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s; each such ultrafilter induces, via binary representations of reals, a set of reals without the Baire property.
[. See in particula]
p. 64
See also
*
*
*
References
{{reflist
External links
Springer Encyclopaedia of Mathematics article on Baire property
Descriptive set theory
Determinacy