In mathematics Baire's space is the set of all infinite number of natural numbers.
This set is the Cartesian product of countless infinity of copies of the set of natural numbers and is usually endowed with the product topology (where each copy of the set of natural numbers is assigned a discrete topology). A space of Baire is a Baire space in the topological sense of the term, and is homomorphic to all of the irrational numbers Ir, to which is assigned the induced topology inherited from the set of real numbers R. The homemorphism between a space of Baire and the set of irrational numbers is constructed using continuous fractions.
A Baire space is often indicated by the B, N, or ω symbols. Moschovakis tells them with N {\displaystyle {\mathcal {N}}} .
B has the same cardinality of R, and sometimes it may be convenient to replace the second with the first one. B is also used in real analysis, where it is considered a uniform space. B and Ir (irrational) uniform structures are, however, different: B is complete as Ir is not. Bibliografiamodifica wikitesto Voices correlateemodify wikitesto
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