If $p$ and $q$ are distinct prime number, are $\mathbb{Q}_p$ and $\mathbb{Q}_q$ homeomorphic as topological space?
2026-04-02 05:21:17.1775107277
Are $\mathbb{Q}_p$ and $\mathbb{Q}_q$ homeomorphic?
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Yes. Let $X$ be the product of countably infinitely many copies of the discrete two-point spaces (i.e., a Cantor set), and let $x\in X$ be any point; then $\Bbb Q_p$ and $\Bbb Q_q$ are both homeomorphic to $X\setminus\{x\}$.
If you prefer, you can start with the middle-thirds Cantor set $C$ and delete any one point: they are homeomorphic to $C\setminus\{1\}$, for instance.