Identification of product of reals and p-adics with $\mathbb{R} / \mathbb{Z}$

Question

Let $p$ be prime. I want to show, that $$\mathbb{Z} \left[ 1/p \right] \setminus \big( \mathbb{R} \times (\mathbb{Q}_p / \mathbb{Z}_p) \big)$$is homeomorphic to $\mathbb{R} / \mathbb{Z}$ via $\big[x,[0] \big] \longmapsto [x]$. By the first quotient I mean the quotient by the diagonal action on $\mathbb{R} \times (\mathbb{Q}_p / \mathbb{Z}_p)$.
Thanks for any hints or clues!

Answer 1

Hint: Show that $$ \left(x,\frac{n}{p^k}+\mathbb{Z}_p\right), \left(x-\frac{n}{p^k},\mathbb{Z}_p\right) $$ are in the same orbit, so the quotient is homeomorphic to $\dfrac{\mathbb{R}}{\operatorname{Stab}_{\mathbb{Z}[1/p]}(\mathbb{Z}_p)}$