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 $\mathbb {Q}$ (with its standard topology).^[1]^[2]^[3]^[4]^[5]^[6]

Examples

As a consequence of the theorem, the metric space $\mathbb {Q} ^{2}$ (with its usual Euclidean distance) is homeomorphic to $\mathbb {Q}$ , which may seem counterintuitive. This is in contrast to, e.g., $\mathbb {R} ^{2}$ , which is not homeomorphic to $\mathbb {R}$ . As another example, $\mathbb {Q} \cap [0,1]$ is also homeomorphic to $\mathbb {Q}$ , again in contrast to the closed real interval $[0,1]$ , which is not homeomorphic to $\mathbb {R}$ (whereas the open interval $(0,1)$ is).

References

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

Examples

References

See also