nLab Baire space of sequences

This entry is about the Baire space of sequences of natural numbers. For another concept of the same name in topology proper, see at Baire space.

Idea

In the study of computability, descriptive set theory, etc, by Baire space is meant the topological space $\mathbb{N}^{\mathbb{N}}$ of infinite sequences of natural numbers equipped with the product topology.

The continuous functions from Baire space to itself serve the role of computable functions in computable analysis. See at computable function (analysis).

The Baire space is homeomorphic to the space of irrational numbers, considered as a subspace of the real numbers equipped with its usual Euclidean topology. See continued fraction for discussion of this point.

Related concepts

• irrational number

• Cantor space

computability

	type I computability	type II computability
typical domain	natural numbers $\mathbb{N}$	Baire space of infinite sequences $\mathbb{B} = \mathbb{N}^{\mathbb{N}}$
computable functions	partial recursive function	computable function (analysis)
type of computable mathematics	recursive mathematics	computable analysis, Type Two Theory of Effectivity
type of realizability	number realizability	function realizability
partial combinatory algebra	Kleene's first partial combinatory algebra	Kleene's second partial combinatory algebra

References

• Tatsuji Kawai, Giovanni Sambin, The principle of pointfree continuity, Logical Methods in Computer Science, Volume 15, Issue 1 (March 5, 2019). (doi:10.23638/LMCS-15%281%3A22%292019, arXiv:1802.04512)

Lecture notes include

• Andrej Bauer, page 5 and section 5.3.1 of Realizability as connection between constructive and computable mathematics, in T. Grubba, P. Hertling, H. Tsuiki, and Klaus Weihrauch, (eds.) CCA 2005 - Second International Conference on Computability and Complexity in Analysis, August 25-29,2005, Kyoto, Japan, ser. Informatik Berichte, , vol. 326-7/2005. FernUniversität Hagen, Germany, 2005, pp. 378–379. (pdf)

Textbook accounts include

• Klaus Weihrauch, section 2.3 of Computable Analysis Berlin: Springer, 2000