Sierpiński's theorem on metric spaces

In mathematics, Sierpiński's theorem is an isomorphism theorem concerning certain metric spaces, named after Wacław Sierpiński who proved it in 1920.[1]

It states that any countable metric space without isolated points is homeomorphic to (with its standard topology).[1][2][3][4][5][6]

Examples

As a consequence of the theorem, the metric space (with its usual Euclidean distance) is homeomorphic to , which may seem counterintuitive. This is in contrast to, e.g., , which is not homeomorphic to . As another example, is also homeomorphic to , again in contrast to the closed real interval , which is not homeomorphic to (whereas the open interval is).

References

  1. ^ a b Sierpiński, Wacław (1920). "Sur une propriété topologique des ensembles dénombrables denses en soi". Fundamenta Mathematicae. 1: 11–16.
  2. ^ Błaszczyk, Aleksander. "A Simple Proof of Sierpiński's Theorem". The American Mathematical Monthly. 126 (5): 464–466. doi:10.1080/00029890.2019.1577103.
  3. ^ Dasgupta, Abhijit. "Countable metric spaces without isolated points" (PDF).
  4. ^ Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. Exercise 6.2.A(d), p. 370. ISBN 3-88538-006-4.
  5. ^ Kechris, Alexander S. (1995). Classical Descriptive Set Theory. Graduate Texts in Mathematics. Springer. Exercise 7.12, p. 40.
  6. ^ van Mill, Jan (2001). The Infinite-Dimensional Topology of Function Spaces. Elsevier. Theorem 1.9.6, p. 76. ISBN 9780080929774.

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