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I am not able to find out what are all the open subgroups of $(\mathbb{R},+)$, open as a set in usual topology and also subgroup.

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Use archimedian property of $\mathbb{R}$

let $S$ be any proper open subgroup, $x\in \mathbb{R}\setminus S, S\le \mathbb{R}$, $0\in S$ is interior point so there exist $\epsilon >0$, such that $(-\epsilon,\epsilon)\subseteq S$ ,forthis $\epsilon>0$ there exists $N\in\mathbb{N}$ such that $\frac{x}{N}<\epsilon$, so $\frac{x}{N}+\dots+\frac{x}{N}(N \text{ times})=x\in S$ so the only subgroups are trivials

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Hint: Every open subgroup of a topological group is also closed.

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HINT: If $G$ is a subgroup of $\Bbb R$, then $0\in G$. If in addition $G$ is open, then $G$ contains an interval of the form $I=(-a,a)$ for some $a>0$. Clearly $G\supseteq nI$ for each $n\in\Bbb Z^+$, where $nI=\{nx:x\in I\}$. What does this tell you about $G$?

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