Baire space (set theory)
Encyclopedia
In set theory
, the Baire space is the set of all infinite sequences of natural number
s with a certain topology
. This space is commonly used in descriptive set theory
, to the extent that its elements are often called “reals.” It is often denoted B, NN, or ωω. Moschovakis
denotes it
.
The Baire space is defined to be the Cartesian product
of countably infinitely
many copies of the set of natural numbers, and is given the product topology
(where each copy of the set of natural numbers is given the discrete topology). The Baire space is often represented using the tree of finite sequences of natural numbers.
The Baire space can be contrasted with Cantor space
, the set of infinite sequences of binary digits.
:
By moving to a different basis for the same topology, an alternate characterization of open sets can be obtained:
Thus a basic open set in the Baire space specifies a finite initial segment τ of an infinite sequence of natural numbers, and all the infinite sequences extending τ form a basic open set. This leads to a representation of the Baire space as the set of all paths through the full tree ω<ω of finite sequences of natural numbers ordered by extension. An open set is determined by some (possibly infinite) union of nodes of the tree; a point in Baire space is in the open set if and only if its path goes through one of these nodes.
The representation of the Baire space as paths through a tree also gives a characterization of closed sets. For any closed subset C of Baire space there is a subtree T of ω<ω such that any point x is in C if and only if x is a path through T. Conversely, the set of paths through any subtree of ω<ω is a closed set.
s when they are given the subspace topology
inherited from the real line. A homeomorphism between Baire space and the irrationals can be constructed using continued fraction
s.
From the point of view of descriptive set theory
, the fact that the real line is connected causes technical difficulties. For this reason, it is more common to study Baire space. Because every Polish space is the continuous image of Baire space, it often possible to prove results about arbitrary Polish spaces by showing these properties hold for Baire space and showing they are preserved by continuous functions.
B is also of independent, but minor, interest in real analysis
, where it is considered as a uniform space
. The uniform structures of B and Ir (the irrationals) are different however: B is complete
in its usual metric while Ir is not (although these spaces are homeomorphic).
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, the Baire space is the set of all infinite sequences of natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s with a certain topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. This space is commonly used in descriptive set theory
Descriptive set theory
In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces...
, to the extent that its elements are often called “reals.” It is often denoted B, NN, or ωω. Moschovakis
Yiannis N. Moschovakis
Yiannis Nicholas Moschovakis is a set theorist, descriptive set theorist, and recursion theorist, at UCLA. For many years he has split his time between UCLA and University of Athens . His book Descriptive Set Theory is the primary reference for the subject...
denotes it
.The Baire space is defined to be the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of countably infinitely
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
many copies of the set of natural numbers, and is given the product topology
Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
(where each copy of the set of natural numbers is given the discrete topology). The Baire space is often represented using the tree of finite sequences of natural numbers.
The Baire space can be contrasted with Cantor space
Cantor space
In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the" Cantor space...
, the set of infinite sequences of binary digits.
Topology and trees
The product topology used to define the Baire space can be described more concretely in terms of trees. The definition of the product topology leads to this characterization of basic open setsBase (topology)
In mathematics, a base B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T...
:
- If any finite set of natural number coordinates {ci : i < n } is selected, and for each ci a particular natural number value vi is selected, then the set of all infinite sequences of natural numbers that have value vi at position ci for all i < n is a basic open set. Every open set is a union of a collection of these.
By moving to a different basis for the same topology, an alternate characterization of open sets can be obtained:
- If a sequence of natural numbers {wi : i < n} is selected, then the set of all infinite sequences of natural numbers that have value wi at position i for all i < n is a basic open set. Every open set is a union of a collection of these.
Thus a basic open set in the Baire space specifies a finite initial segment τ of an infinite sequence of natural numbers, and all the infinite sequences extending τ form a basic open set. This leads to a representation of the Baire space as the set of all paths through the full tree ω<ω of finite sequences of natural numbers ordered by extension. An open set is determined by some (possibly infinite) union of nodes of the tree; a point in Baire space is in the open set if and only if its path goes through one of these nodes.
The representation of the Baire space as paths through a tree also gives a characterization of closed sets. For any closed subset C of Baire space there is a subtree T of ω<ω such that any point x is in C if and only if x is a path through T. Conversely, the set of paths through any subtree of ω<ω is a closed set.
Properties
The Baire space has the following properties:- It is a perfect Polish spacePolish spaceIn 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...
, which means it is a completely metrizable second countable space with no isolated pointIsolated pointIn topology, a branch of mathematics, a point x of a set S is called an isolated point of S, if there exists a neighborhood of x not containing other points of S.In particular, in a Euclidean space ,...
s. As such, it has the same cardinality as the real line and is a Baire spaceBaire spaceIn mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
in the topological sense of the term. - It is zero dimensional and totally disconnected.
- It is not locally compact.
- It is universal for Polish spaces in the sense that it can be mapped continuously onto any non-empty Polish space.
- The Baire space is homeomorphic to the product of any finite or countable number of copies of itself.
Relation to the real line
The Baire space is homeomorphic to the set of irrational numberIrrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
s when they are given the subspace topology
Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
inherited from the real line. A homeomorphism between Baire space and the irrationals can be constructed using continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
s.
From the point of view of descriptive set theory
Descriptive set theory
In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces...
, the fact that the real line is connected causes technical difficulties. For this reason, it is more common to study Baire space. Because every Polish space is the continuous image of Baire space, it often possible to prove results about arbitrary Polish spaces by showing these properties hold for Baire space and showing they are preserved by continuous functions.
B is also of independent, but minor, interest in real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
, where it is considered as a uniform space
Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
. The uniform structures of B and Ir (the irrationals) are different however: B is complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
in its usual metric while Ir is not (although these spaces are homeomorphic).