Is Z discrete in its profinite completion?

Question

Consider the inclusion $\phi: \mathbb{Z} \to \hat{\mathbb{Z}}$ where $\hat{\mathbb{Z}} : = \varprojlim_n\mathbb{Z}/n\mathbb{Z}$. Is the subspace topology on $\mathbb{Z}$ from this inclusion the discrete topology? Is this true for all $G \to \hat{G}$?

Answer 1

No, the image of $Z$ is dense in its profinite completion which is a compact group. https://en.wikipedia.org/wiki/Profinite_group#Profinite_completion

You can't have an infinite discrete subset in a compact space.

Answer 2

No, $n!\to0$ in the profinite topology.