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.Template:R

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

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., ${ℝ}^2$, which is not homeomorphic to $ℝ$. 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 $ℝ$ (whereas the open interval $(0,1)$ is).

See also

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