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In set theory, the Baire space is the set of all infinite sequences of natural numbers 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 denoted B, NN, ωω, ωω, or . 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. == Topology and trees == The product topology used to define the Baire space can be described more concretely in terms of trees. The basic open sets of the product topology are cylinder sets, here characterized as: :If any finite set of natural number coordinates is selected, and for each ''c''''i'' a particular natural number value ''v''''i'' is selected, then the set of all infinite sequences of natural numbers that have value ''v''''i'' at position ''c''''i'' 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 is selected, then the set of all infinite sequences of natural numbers that have value ''w''''i'' 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Baire space (set theory)」の詳細全文を読む スポンサード リンク
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