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?