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General topology Cantor sets and descriptive spaces Object

Baire space of sequences

Builds on Product topologyNatural numbersComplete metric spaceSeparable spaceBaire spaceIrrational number

Description

The Baire space ℕℕ\N^{\N} is the set of all infinite sequences of natural numbers. Its topology is the product topology of countably many copies of the discrete space ℕ\N. Its basic open sets are the sets of all sequences that begin with a given finite sequence. The topology comes from the metric d(x,y)=2−nd(x, y) = 2^{-n}, where nn is the first index with xn≠ynx_n \neq y_n, and d(x,x)=0d(x, x) = 0. With this metric the space is complete, and it is separable, since the eventually zero sequences are dense. In descriptive set theory it is written 𝒩\mathcal{N} or ωω\omega^\omega.

Remarks

The Baire space is homeomorphic to the set of irrational numbers with the topology of the real line; one homeomorphism uses continued fraction expansions. It is not locally compact: no basic open set is compact, since the next term of the sequence can take infinitely many values. It has a base of clopen sets. Every nonempty Polish space, that is a separable completely metrizable space, is a continuous image of ℕℕ\N^{\N}. As a complete metric space it is a Baire space in the sense of Baire category; the two uses of the name are related but different.

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Baire space, concept, no evidence levelBaire spaceconceptComplete metric space, concept, no evidence levelComplete metric spaceconceptIrrational number, concept, no evidence levelIrrational numberconceptNatural numbers, object, no evidence levelNatural numbersobjectProduct topology, concept, no evidence levelProduct topologyconceptSeparable space, concept, no evidence levelSeparable spaceconceptBaire space of sequences, object, no evidence levelBaire space of sequen…object
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  1. Baire space concept
  2. Complete metric space concept
  3. Irrational number concept
  4. Natural numbers object
  5. Product topology concept
  6. Separable space concept
  7. Baire space of sequences object

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Further reading

  1. Web https://en.wikipedia.org/wiki/Baire_space_(set_theory)
About this page
TagP2WS16HKD7
URIm2://P2WS16HKD7
Kindobject
WritingW0 stub
MSC54H05, 03E15
UsesProduct topology, Natural numbers, Complete metric space, Separable space, Baire space, Irrational number
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Changed2 November 2026

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