Let $ H\subseteq G$ be a subgroup of a topological group $ G$ (henceforth abbreviated “group”). If the induced topology on $ H$ is discrete, then we say that $ H$ is a discrete subgroup of $ G$. A commonplace example is the subgroup $ \mathbb{Z}\subseteq \mathbb{R}$: the integers are normal subgroup of the real numbers (with the standard topology).

Observe that in $ \mathbb{R}$, the subset $ \mathbb{Z}$ is also a closed set. What about an arbitrary discrete subgroup $ H\subseteq\mathbb{R}$? In other words, if $ H$ is discrete, is $ H$ necessarily closed?

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