Why $\mathbb{Z}$ with $p$-adic topology is non-discrete?
Note1 : discrete : each singleton is an open set.
Note2 : Let the topology $\tau$ on $\mathbb{Z}_p$ be defined by taking as a basis all sets of the form $\{n + \lambda p^a \ ; \ \lambda \in \mathbb{Z}_p \& \ a \in \mathbb{N} \}$. Then $\mathbb{Z}_p$ is a compactification of $\mathbb{Z}$, under the derived topology (it is not a compactification of $\mathbb{Z}$ with its usual topology). The relative topology on $\mathbb{Z}$ as a subset of $\mathbb{Z}_p$ is called the $p$-adic topology on $\mathbb{Z}$.