{"history":[],"article":{"id":"145dd39d-0523-47fe-abfa-70090c511c6f","title":"Baire space (set theory)","slug":"baire-space-set-theory","content":":::summary\n:::\n\n# \nIn [set theory](https://bliptext.com/articles/[set](https://bliptext.com/articles/set-mathematics)-theory), the **Baire space** is the [set](https://bliptext.com/articles/set-mathematics) of all [infinite sequences](https://bliptext.com/articles/infinite-sequence) of [natural numbers](https://bliptext.com/articles/natural-number) with a certain [topology](https://bliptext.com/articles/topology), called the product [topology](https://bliptext.com/articles/topology).\nThis space is commonly used in [descriptive set theory](https://bliptext.com/articles/descriptive-set-theory), to the extent that its elements are often called \"reals\".\nIt is denoted by \\N^{\\N}, or ωω, or by the symbol \\mathcal{N} or sometimes by ωω (not to be confused with the countable ordinal obtained by [ordinal exponentiation](https://bliptext.com/articles/ordinal-arithmetic)).\nThe Baire space is defined to be the [Cartesian product](https://bliptext.com/articles/cartesian-product) of [countably infinitely](https://bliptext.com/articles/countable-set) 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](https://bliptext.com/articles/discrete-topology)).\nThe Baire space is often represented using the [tree](https://bliptext.com/articles/tree-descriptive-set-theory) of finite sequences of natural numbers.\n(This sp**a**ce should **a**lso not be confused with the concept of **a** B**a**ire sp**a**ce, which is **a** cert**a**in kind of topologic**a**l sp**a**ce.)\nThe Baire space can be contrasted with [Cantor space](https://bliptext.com/articles/cantor-space), the set of infinite sequences of [binary digits](https://bliptext.com/articles/binary-digit).\n\n# Topology and trees\nThe [product topology](https://bliptext.com/articles/product-topology) used to define the Baire space can be described in one of two equivalent ways: in terms of a [basis](https://bliptext.com/articles/base-topology) consisting of cylinder sets, or of a [basis](https://bliptext.com/articles/base-topology) of trees.\n\n## Cylinder set basis\nThe [basic open sets](https://bliptext.com/articles/base-topology) of the product topology are [cylinder sets](https://bliptext.com/articles/cylinder-set).\nThese can be characterized as:\nUsing more formal notation, one can define the individual cylinders as\nfor a fixed integer location n and integer value v.\nThe cylinders are then the generators for the cylinder sets: the cylinder sets then consist of all intersections of a finite number of cylinders.\nThat is, given any finite set of natural number coordinates and corresponding natural number values v_i for each i\\in I, one considers the finite intersection of cylinders\nThis intersection is called a [cylinder set](https://bliptext.com/articles/cylinder-set), and the set of all such [cylinder set](https://bliptext.com/articles/cylinder-set)s provides a basis for the [product topology](https://bliptext.com/articles/product-topology).\nEvery open set is a countable union of such cylinder sets.\n\n## Tree basis\nAn alternative basis for the product topology can be given in terms of trees.\nThe basic open sets can be characterized as:\nThus a basic open set in the Baire space is the set of all infinite sequences of natural numbers extending a common finite initial segment σ.\nThis leads to a representation of the Baire space as the set of all infinite paths passing through the full tree ω<ω of finite sequences of natural numbers [ordered by extension](https://bliptext.com/articles/shortlex).\nEach finite initial segment σ is a **node** of the tree of finite sequences.\nEach open set is determined by a countable union S of nodes of that tree.\nA point in Baire space is in an open set if and only if its path goes through one of the nodes in its determining union.\nConversely, each open set corresponds to a subtree S of the full tree ω<ω, consisting of at most a countable number of nodes.\nThe representation of the Baire space as paths through a tree also gives a characterization of closed sets as complements of subtrees defining the open sets.\nEvery point in Baire space passes through a sequence of nodes of ω<ω.\nClosed sets are complements of open sets.\nThis defines a subtree T of the full tree ω<ω, in which the nodes of S defining the open set are missing.\nThe subtree T consists of all nodes in ω<ω that are not in S.\nThis subtree T defines a closed subset C of Baire space such that any point x is in C if and only if x is a path through T.\nConversely, for any closed subset C of Baire space there is a subtree T which consists of all of ω<ω with at most a countable number of nodes removed.\nSince the full tree ω<ω is itself countable, this implies the closed sets correspond to any subtree of the full tree, including finite subtrees.\nThus, the topology consists of [clopen sets](https://bliptext.com/articles/clopen-set).\nThis implies that the Baire space is [zero-dimensional](https://bliptext.com/articles/zero-dimensional) with respect to the small [inductive dimension](https://bliptext.com/articles/inductive-dimension) (as are all spaces whose base consists of clopen sets.)\nThe above definitions of open and closed sets provide the first two sets and of the [boldface](https://bliptext.com/articles/pointclass) [Borel hierarchy](https://bliptext.com/articles/borel-hierarchy).\n\n## Box topology\nCartesian products also have an alternate topology, the [box topology](https://bliptext.com/articles/box-topology).\nThis topology is much finer than the product topology as it does not limit the indicator set to be finite.\nConventionally, Baire space does not refer to this topology; it only refers to the product topology.\n\n# Weight\nThe above definition of the Baire space generalizes to one where the elements x_i of the countably infinite sequence are chosen from a set D(\\kappa) of [cardinality](https://bliptext.com/articles/cardinality) \\kappa.\nSuch a space is called a **Baire space of weight \\kappa** and can be denoted as B(\\kappa).\nWith this definition, the Baire spaces of finite weight would correspond to the [Cantor space](https://bliptext.com/articles/cantor-space).\nThe first Baire space of infinite weight is then B(\\aleph_0); it is homeomorphic to defined above.\n\n# Metric\nGiven two sequences and, a [metric](https://bliptext.com/articles/metric-space) \\rho(x,y) may be defined as where k is the least integer such that x_k\\ne y_k.\nWith this metric, the basic open sets of the tree basis are [balls](https://bliptext.com/articles/ball-mathematics) of radius 1/k.\nA [metric space](https://bliptext.com/articles/metric-space) X [embeds](https://bliptext.com/articles/embedding) into the Baire space B(\\kappa) if and only if X poses a [base](https://bliptext.com/articles/base-topology) \\mathcal{B} of clopen sets, where the cardinality of \\mathcal{B} is less than or equal to \\kappa.\n\n# Properties\nThe Baire space has the following properties:\n\n# Relation to the real line\nThe Baire space is [homeomorphic](https://bliptext.com/articles/homeomorphic) to the set of [irrational numbers](https://bliptext.com/articles/irrational-number) when they are given the [subspace topology](https://bliptext.com/articles/subspace-topology) inherited from the real line.\nA homeomorphism between Baire space and the irrationals can be constructed using [continued fractions](https://bliptext.com/articles/continued-fraction).\nThat is, given a sequence of [natural numbers](https://bliptext.com/articles/natural-number), we can assign a corresponding irrational number greater than 1\nUsing we get another homeomorphism from to the irrationals in the open unit interval (0,1) and we can do the same for the negative irrationals.\nWe see that the irrationals are the topological sum of four spaces homeomorphic to the Baire space and therefore also homeomorphic to the Baire space.\nFrom the point of view of [descriptive set theory](https://bliptext.com/articles/descriptive-set-theory), Baire spaces are more flexible than the [real line](https://bliptext.com/articles/real-line) in the following sense.\nBecause the [real line](https://bliptext.com/articles/real-line) is path-connected, so is every continuous image of a [real line](https://bliptext.com/articles/real-line).\nIn contrast, every [Polish space](https://bliptext.com/articles/polish-space) is the continuous image of Baire space.\nThis difference makes the real line \"slightly awkward to use\", despite the focus of descriptive set theory on sets of reals.\nInstead, it is often possible to prove results about arbitrary Polish spaces by showing that these properties hold for Baire space and are preserved by [continuous functions](https://bliptext.com/articles/continuous-functions).\nωω is also of independent, but minor, interest in [real analysis](https://bliptext.com/articles/real-analysis), where it is considered as a [uniform space](https://bliptext.com/articles/uniform-space).\nThe uniform structures of ωω and ****Ir**** (the irrationals) are different, however: ωω is [complete](https://bliptext.com/articles/complete-space) in its usual metric while ****Ir**** is not (although these spaces are homeomorphic).\n\n# The shift operator\nThe [shift operator](https://bliptext.com/articles/shift-operator) on Baire space, when mapped to the [unit interval](https://bliptext.com/articles/unit-interval) of the [reals](https://bliptext.com/articles/real-numbers), becomes the [Gauss–Kuzmin–Wirsing operator](https://bliptext.com/articles/gauss-kuzmin-wirsing-operator).\nThat is, given a [sequence](https://bliptext.com/articles/sequence), the shift operator T returns.\nLikewise, given the continued fraction, the Gauss map returns.\nThe corresponding operator for functions from Baire space to the complex plane is the [Gauss–Kuzmin–Wirsing operator](https://bliptext.com/articles/gauss-kuzmin-wirsing-operator); it is the [transfer operator](https://bliptext.com/articles/transfer-operator) of the Gauss map.\nThat is, one considers maps from Baire space to the [complex plane](https://bliptext.com/articles/complex-plane) \\Complex.\nThis space of maps inherits a topology from the product topology on Baire space; for example, one may consider functions having [uniform convergence](https://bliptext.com/articles/uniform-convergence).\nThe shift map, acting on this space of functions, is then the GKW operator.\nThe [Haar measure](https://bliptext.com/articles/haar-measure) of the shift operator, that is, a function that is invariant under shifts, is given by the [Minkowski measure](https://bliptext.com/articles/minkowski-s-question-mark-function) (...)'.\nThat is, one has that (TE)' = E', where T is the shift and E any [measurable subset](https://bliptext.com/articles/measurable-set) of ωω.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T21:42:49.978Z","updated_at":"2025-01-25T21:42:49.978Z","search_vector":"'bair':1 'set':3 'space':2 'theori':4","search_content":" \n \n\n# \nIn [[set](https://bliptext.com/articles/set-mathematics) theory](https://bliptext.com/articles/[set](https://bliptext.com/articles/set-mathematics)-theory), the **Baire space** is the [set](https://bliptext.com/articles/set-mathematics) of all [infinite sequences](https://bliptext.com/articles/infinite-sequence) of [natural numbers](https://bliptext.com/articles/natural-number) with a certain [topology](https://bliptext.com/articles/topology), called the product [topology](https://bliptext.com/articles/topology).\nThis space is commonly used in [descriptive set theory](https://bliptext.com/articles/descriptive-set-theory), to the extent that its elements are often called \"reals\".\nIt is denoted by \\N^{\\N}, or ωω, or by the symbol \\mathcal{N} or sometimes by ωω (not to be confused with the countable ordinal obtained by [ordinal exponentiation](https://bliptext.com/articles/ordinal-arithmetic)).\nThe Baire space is defined to be the [Cartesian product](https://bliptext.com/articles/cartesian-product) of [countably infinitely](https://bliptext.com/articles/countable-set) 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](https://bliptext.com/articles/discrete-topology)).\nThe Baire space is often represented using the [tree](https://bliptext.com/articles/tree-descriptive-set-theory) of finite sequences of natural numbers.\n(This sp**a**ce should **a**lso not be confused with the concept of **a** B**a**ire sp**a**ce, which is **a** cert**a**in kind of topologic**a**l sp**a**ce.)\nThe Baire space can be contrasted with [Cantor space](https://bliptext.com/articles/cantor-space), the set of infinite sequences of [binary digits](https://bliptext.com/articles/binary-digit).\n\n# Topology and trees\nThe [product topology](https://bliptext.com/articles/product-topology) used to define the Baire space can be described in one of two equivalent ways: in terms of a [basis](https://bliptext.com/articles/base-topology) consisting of cylinder sets, or of a [basis](https://bliptext.com/articles/base-topology) of trees.\n\n## Cylinder set basis\nThe [basic open sets](https://bliptext.com/articles/base-topology) of the product topology are [cylinder sets](https://bliptext.com/articles/cylinder-set).\nThese can be characterized as:\nUsing more formal notation, one can define the individual cylinders as\nfor a fixed integer location n and integer value v.\nThe cylinders are then the generators for the cylinder sets: the cylinder sets then consist of all intersections of a finite number of cylinders.\nThat is, given any finite set of natural number coordinates and corresponding natural number values v_i for each i\\in I, one considers the finite intersection of cylinders\nThis intersection is called a [cylinder set](https://bliptext.com/articles/cylinder-set), and the set of all such [cylinder set](https://bliptext.com/articles/cylinder-set)s provides a basis for the [product topology](https://bliptext.com/articles/product-topology).\nEvery open set is a countable union of such cylinder sets.\n\n## Tree basis\nAn alternative basis for the product topology can be given in terms of trees.\nThe basic open sets can be characterized as:\nThus a basic open set in the Baire space is the set of all infinite sequences of natural numbers extending a common finite initial segment σ.\nThis leads to a representation of the Baire space as the set of all infinite paths passing through the full tree ω<ω of finite sequences of natural numbers [ordered by extension](https://bliptext.com/articles/shortlex).\nEach finite initial segment σ is a **node** of the tree of finite sequences.\nEach open set is determined by a countable union S of nodes of that tree.\nA point in Baire space is in an open set if and only if its path goes through one of the nodes in its determining union.\nConversely, each open set corresponds to a subtree S of the full tree ω<ω, consisting of at most a countable number of nodes.\nThe representation of the Baire space as paths through a tree also gives a characterization of closed sets as complements of subtrees defining the open sets.\nEvery point in Baire space passes through a sequence of nodes of ω<ω.\nClosed sets are complements of open sets.\nThis defines a subtree T of the full tree ω<ω, in which the nodes of S defining the open set are missing.\nThe subtree T consists of all nodes in ω<ω that are not in S.\nThis subtree T defines a closed subset C of Baire space such that any point x is in C if and only if x is a path through T.\nConversely, for any closed subset C of Baire space there is a subtree T which consists of all of ω<ω with at most a countable number of nodes removed.\nSince the full tree ω<ω is itself countable, this implies the closed sets correspond to any subtree of the full tree, including finite subtrees.\nThus, the topology consists of [clopen sets](https://bliptext.com/articles/clopen-set).\nThis implies that the Baire space is [zero-dimensional](https://bliptext.com/articles/zero-dimensional) with respect to the small [inductive dimension](https://bliptext.com/articles/inductive-dimension) (as are all spaces whose base consists of clopen sets.)\nThe above definitions of open and closed sets provide the first two sets and of the [boldface](https://bliptext.com/articles/pointclass) [Borel hierarchy](https://bliptext.com/articles/borel-hierarchy).\n\n## Box topology\nCartesian products also have an alternate topology, the [box topology](https://bliptext.com/articles/box-topology).\nThis topology is much finer than the product topology as it does not limit the indicator set to be finite.\nConventionally, Baire space does not refer to this topology; it only refers to the product topology.\n\n# Weight\nThe above definition of the Baire space generalizes to one where the elements x_i of the countably infinite sequence are chosen from a set D(\\kappa) of [cardinality](https://bliptext.com/articles/cardinality) \\kappa.\nSuch a space is called a **Baire space of weight \\kappa** and can be denoted as B(\\kappa).\nWith this definition, the Baire spaces of finite weight would correspond to the [Cantor space](https://bliptext.com/articles/cantor-space).\nThe first Baire space of infinite weight is then B(\\aleph_0); it is homeomorphic to defined above.\n\n# Metric\nGiven two sequences and, a [metric](https://bliptext.com/articles/metric-space) \\rho(x,y) may be defined as where k is the least integer such that x_k\\ne y_k.\nWith this metric, the basic open sets of the tree basis are [balls](https://bliptext.com/articles/ball-mathematics) of radius 1/k.\nA [metric space](https://bliptext.com/articles/metric-space) X [embeds](https://bliptext.com/articles/embedding) into the Baire space B(\\kappa) if and only if X poses a [base](https://bliptext.com/articles/base-topology) \\mathcal{B} of clopen sets, where the cardinality of \\mathcal{B} is less than or equal to \\kappa.\n\n# Properties\nThe Baire space has the following properties:\n\n# Relation to the real line\nThe Baire space is [homeomorphic](https://bliptext.com/articles/homeomorphic) to the set of [irrational numbers](https://bliptext.com/articles/irrational-number) when they are given the [subspace topology](https://bliptext.com/articles/subspace-topology) inherited from the real line.\nA homeomorphism between Baire space and the irrationals can be constructed using [continued fractions](https://bliptext.com/articles/continued-fraction).\nThat is, given a sequence of [natural numbers](https://bliptext.com/articles/natural-number), we can assign a corresponding irrational number greater than 1\nUsing we get another homeomorphism from to the irrationals in the open unit interval (0,1) and we can do the same for the negative irrationals.\nWe see that the irrationals are the topological sum of four spaces homeomorphic to the Baire space and therefore also homeomorphic to the Baire space.\nFrom the point of view of [descriptive set theory](https://bliptext.com/articles/descriptive-set-theory), Baire spaces are more flexible than the [real line](https://bliptext.com/articles/real-line) in the following sense.\nBecause the [real line](https://bliptext.com/articles/real-line) is path-connected, so is every continuous image of a [real line](https://bliptext.com/articles/real-line).\nIn contrast, every [Polish space](https://bliptext.com/articles/polish-space) is the continuous image of Baire space.\nThis difference makes the real line \"slightly awkward to use\", despite the focus of descriptive set theory on sets of reals.\nInstead, it is often possible to prove results about arbitrary Polish spaces by showing that these properties hold for Baire space and are preserved by [continuous functions](https://bliptext.com/articles/continuous-functions).\nωω is also of independent, but minor, interest in [real analysis](https://bliptext.com/articles/real-analysis), where it is considered as a [uniform space](https://bliptext.com/articles/uniform-space).\nThe uniform structures of ωω and ****Ir**** (the irrationals) are different, however: ωω is [complete](https://bliptext.com/articles/complete-space) in its usual metric while ****Ir**** is not (although these spaces are homeomorphic).\n\n# The shift operator\nThe [shift operator](https://bliptext.com/articles/shift-operator) on Baire space, when mapped to the [unit interval](https://bliptext.com/articles/unit-interval) of the [reals](https://bliptext.com/articles/real-numbers), becomes the [Gauss–Kuzmin–Wirsing operator](https://bliptext.com/articles/gauss-kuzmin-wirsing-operator).\nThat is, given a [sequence](https://bliptext.com/articles/sequence), the shift operator T returns.\nLikewise, given the continued fraction, the Gauss map returns.\nThe corresponding operator for functions from Baire space to the complex plane is the [Gauss–Kuzmin–Wirsing operator](https://bliptext.com/articles/gauss-kuzmin-wirsing-operator); it is the [transfer operator](https://bliptext.com/articles/transfer-operator) of the Gauss map.\nThat is, one considers maps from Baire space to the [complex plane](https://bliptext.com/articles/complex-plane) \\Complex.\nThis space of maps inherits a topology from the product topology on Baire space; for example, one may consider functions having [uniform convergence](https://bliptext.com/articles/uniform-convergence).\nThe shift map, acting on this space of functions, is then the GKW operator.\nThe [Haar measure](https://bliptext.com/articles/haar-measure) of the shift operator, that is, a function that is invariant under shifts, is given by the [Minkowski measure](https://bliptext.com/articles/minkowski-s-question-mark-function) (...)'.\nThat is, one has that (TE)' = E', where T is the shift and E any [measurable subset](https://bliptext.com/articles/measurable-set) of ωω.","revisionCount":0},"hasMore":false,"nextCursor":null}

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