Relation To The Real Line
The Baire space is homeomorphic to the set of irrational numbers when they are given the subspace topology inherited from the real line. A homeomorphism between Baire space and the irrationals can be constructed using continued fractions.
From the point of view of descriptive set theory, the fact that the real line is connected causes technical difficulties. For this reason, it is more common to study Baire space. Because every Polish space is the continuous image of Baire space, it is often possible to prove results about arbitrary Polish spaces by showing that these properties hold for Baire space and by showing that they are preserved by continuous functions.
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 (the irrationals) are different, however: B is complete in its usual metric while Ir is not (although these spaces are homeomorphic).
Read more about this topic: Baire Space (set Theory)
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