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

It states that any countable metric space without isolated points is homeomorphic to ${\displaystyle \mathbb {Q} }$ (with its standard topology).

Examples

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

References

1. ^ 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.

See also

• Cantor's isomorphism theorem is an analogous statement on linear orders.

• Theorems in topology