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General topology Cantor sets and descriptive spaces Object

Cantor space

Builds on Product topologyCompact spaceTotally disconnected spaceCantor setCantor set is homeomorphic to the space of binary sequencesMetrizable space

Description

The Cantor space is the set 2ℕ={0,1}ℕ2^{\N} = \set{0, 1}^{\N} of all infinite sequences of 00s and 11s, with the product topology of the discrete space {0,1}\set{0, 1}. Its basic open sets are the sets of all sequences that begin with a given finite word. 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. The space is compact by Tychonoff's theorem, totally disconnected, and has no isolated points. More generally, a topological space is called a Cantor space when it is homeomorphic to 2ℕ2^{\N}.

Remarks

The Cantor set is a Cantor space: it is homeomorphic to the space of binary sequences. By Brouwer's theorem, a space is a Cantor space if and only if it is nonempty, compact, metrizable, totally disconnected and without isolated points. For this reason 2ℕ×2ℕ2^{\N} \times 2^{\N} and (2ℕ)ℕ(2^{\N})^{\N} are again Cantor spaces. Every nonempty compact metric space is a continuous image of the Cantor space.

Reading path

To understand this page from the ground up, read these 63 pages in order. The foundations come first.

  1. Cantor set example
  2. Compact space concept
  3. Element definition
  4. Metrizable space concept
  5. Product topology concept
  6. Proposition concept
  7. Real numbers object
  8. Totally disconnected space concept
  9. Set definition
  10. Conjunction definition
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Empty set object
  15. Family of sets definition
  16. Ordered pair definition
  17. Intersection definition
  18. Union definition
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. Cartesian product definition
  22. Subset definition
  23. Intersection of a family of sets definition
  24. Union of a family of sets definition
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Universal set definition
  28. Topological space definition
  29. Complement definition
  30. Function definition
  31. Subspace topology definition
  32. Basis of a topology definition
  33. Codomain definition
  34. Injective function definition
  35. Preimage of a set definition
  36. Product topology definition
  37. Metric space definition
  38. Subbasis definition
  39. Surjective function definition
  40. Product topology on an arbitrary product definition
  41. Open ball definition
  42. Bijective function definition
  43. Metric topology definition
  44. Inverse function definition
  45. Open set definition
  46. Closed set definition
  47. Hausdorff space definition
  48. Continuous map of topological spaces definition
  49. Open cover definition
  50. Complement of an open set is closed statement
  51. Homeomorphism definition
  52. Compact space definition
  53. Continuity via closed sets statement
  54. Compact set definition
  55. Compactness via open covers statement
  56. Closed subsets of compact sets are compact statement
  57. Compact subsets of a Hausdorff space are closed statement
  58. Continuous image of a compact set is compact statement
  59. Product of compact sets is compact statement
  60. Continuous bijection from a compact space to a Hausdorff space is a homeomorphism statement
  61. Tychonoff's theorem statement
  62. Cantor set is homeomorphic to the space of binary sequences statement
  63. Cantor space object

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cantor_space
About this page
TagW9VYJRBF9N
URIm2://W9VYJRBF9N
Kindobject
WritingW0 stub
MSC54F65, 54H05
UsesProduct topology, Compact space, Totally disconnected space, Cantor set, Cantor set is homeomorphic to the space of binary sequences, Metrizable space
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Changed2 November 2026

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