## Flipping between p-adics and reals

Let $x\in \mathbb{Q}_p$. We can represent $x$ by the p-adic series $x=\sum_{n\ge k}a_np^n, k,a_n \in \mathbb{Z}, 0\le a_n < p$. Define $f_p:\mathbb{Q}_p \to \mathbb{R}$ by $f_p(x) = \sum_{n \le -k}a_{-n} p^{-n-1}$ (essentially flipping the digits across the decimal point).

Define the relation $\sim$ by $x,y \in \mathbb{Q}_p, x\sim y$ if $f_p(x) = f_p(y)$. Is the quotient topology for $\mathbb{Q}_p / \sim$ metrizable?