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

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, ${\displaystyle \mathbb {Q} \cap [0,1]}$ is also homeomorphic to ${\displaystyle \mathbb {Q} }$, again in contrast to the closed real interval ${\displaystyle [0,1]}$, which is not homeomorphic to ${\displaystyle \mathbb {R} }$ (whereas the open interval ${\displaystyle (0,1)}$ is).