About: Baire space (set theory)

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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 NN, ωω, by the symbol or also ωω, not to be confused with the countable ordinal obtained by ordinal exponentiation. The Baire space can be contrasted with Cantor space, the set of infinite sequences of binary digits.

Attributes	Values
rdf:type	Thing yago:Abstraction100002137 yago:Possession100032613 yago:Property113244109 yago:Relation100031921 yago:WikicatPropertiesOfTopologicalSpaces
rdfs:label	فضاء بير (نظرية المجموعات) (ar) Baire-Raum (speziell) (de) Baire space (set theory) (en) Spazio di Baire (teoria degli insiemi) (it) Espace de Baire (théorie des ensembles) (fr) ベール空間 (集合論) (ja) Baire-ruimte (verzamelingenleer) (nl)
rdfs:comment	نسمي فضاء طبولوجيا فضاء بير Baire space إذا كانت كل التقاطعات في مجموعة منتهية لمفتوح كثيف كثيفة، وذلك على اسم رينيه-لوي بير الذي قدم هذا المفهوم. (ar) En mathématiques, et plus précisément en topologie générale, l’espace de Baire est le nom donné — d'après René Baire — à l'ensemble de toutes les suites d'entiers, muni d'une certaine topologie. Cet espace est souvent utilisé en théorie descriptive des ensembles, au point que ses éléments sont souvent appelés des « réels ». On le note souvent B, NN, ωω, ou ωω. (fr) 集合論におけるベール空間（ベールくうかん、英: Baire space）は、各項が自然数に値をとるすべてからなる集合に適当な位相を入れたものである。この空間は、その元を以て「実数」と称される程、記述集合論でよく用いられる。ベール空間を表す記号としては B, NN や ωω がよく用いられる。はと書いている。ベール空間は自然数全体の成す集合 ω の可算無限個のコピーのデカルト積に積位相を入れたもの（ただし ω の各コピーには離散位相が入る）として定義することができる。ベール空間は、自然数からなる全ての有限列の成すとして表されることが多い。ベール空間はカントル空間（全ての二進無限列からなる集合）と対比される。 (ja) 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 NN, ωω, by the symbol or also ωω, not to be confused with the countable ordinal obtained by ordinal exponentiation. The Baire space can be contrasted with Cantor space, the set of infinite sequences of binary digits. (en) Der Baire-Raum (nach dem französischen Mathematiker René Louis Baire) ist ein topologischer Raum. Seine Grundmenge ist die Menge aller unendlichenFolgen von natürlichen Zahlen, und die Topologie des Baire-Raums (die Menge der offenen Mengen des Baire-Raums) besteht aus denjenigen Mengen solcher Folgen, die die Vereinigung von Mengen aus Folgen mit gemeinsamem Präfix sind. (de) In matematica lo spazio di Baire è l'insieme di tutte le successioni infinite di numeri naturali. Tale insieme è il prodotto cartesiano di un'infinità numerabile di copie dell'insieme dei numeri naturali ed è usualmente dotato della topologia prodotto (ove ad ogni copia dell'insieme dei numeri naturali è assegnata la topologia discreta). Uno spazio di Baire è uno spazio di Baire nel senso topologico del termine, ed è omeomorfo all'insieme dei numeri irrazionali Ir a cui è assegnata la topologia indotta ereditata dall'insieme dei numeri reali R. L'omeomorfismo tra uno spazio di Baire e l'insieme dei numeri irrazionali si costruisce utilizzando le frazioni continue. (it) In de verzamelingenleer, een deelgebied van de wiskunde, is de baire-ruimte de verzameling van alle oneindige rijen van natuurlijke getallen uitgerust met een bepaalde topologie. Deze ruimte wordt vaak gebruikt in de beschrijvende verzamelingenleer. De baire-ruimte wordt vaak aangeduid met of . duidt de ruimte aan met . De baire-ruimte kan worden vergeleken met de cantor-ruimte, de verzameling van oneindige rijen van binaire cijfers. (nl)
rdfs:seeAlso	Countable set
dcterms:subject	Descriptive set theory Topological spaces Integer sequences
Wikipage page ID	2078245 (xsd:integer)
Wikipage revision ID	1106565862 (xsd:integer)
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sameAs	Baire space (set theory) Baire space (set theory) Baire space (set theory) Baire space (set theory) Baire space (set theory) Baire space (set theory) Baire space (set theory) Baire space (set theory) Baire space (set theory) Baire space (set theory) Baire space (set theory)
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has abstract	نسمي فضاء طبولوجيا فضاء بير Baire space إذا كانت كل التقاطعات في مجموعة منتهية لمفتوح كثيف كثيفة، وذلك على اسم رينيه-لوي بير الذي قدم هذا المفهوم. (ar)

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