In mathematics, the Baire space is the set of all infinite sequences of natural numbers. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... This is a page about mathematics. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), and they can be used for ordering (this is...

As usually represented, this is equal to 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). Baire space is a Baire space in the topological sense, and is homeomorphic to the set of irrational numbers Ir, with its subspace topology inherited from the real numbers R. The homeomorphism between Baire space and the irrationals can be constructed using continued fractions. In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are enough points for certain limit processes. ... This word should not be confused with homomorphism. ... In mathematics, an irrational number is any real number that is not a rational number, i. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

Baire space is often denoted B, N^N, or ω^ω. Moschovakis denotes it .

B has the same cardinality as R, and can be used as a convenient substitute for R in some set-theoretical contexts. In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ...

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 are different however: B is complete and Ir is not. Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... In the mathematical field of topology, a uniform space is a set with a uniform structure. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...

Baire space should be contrasted with Cantor space, the set of infinite sequences of binary digits. In mathematics, the term Cantor space is sometimes used to denote the topological abstraction of the classical Cantor set: A topological space is a Cantor space if it is homeomorphic to the Cantor set. ... This article is about the unit of information, see Bit (disambiguation) for other meanings. ...

References

• Moschovakis, Yiannis N. (1980). Descriptive Set Theory, North Holland. ISBN 0-444-70199-0.