Open subgroup

Remarks

Example: in $\R \times \Z$ with the product topology, where $\Z$ is discrete, the subgroup $\R \times \set{0}$ is open, because $\set{0}$ is open in $\Z$. Non-example: $\Z$ is a subgroup of $(\R, +)$ but it is not open, since no interval around $0$ lies in $\Z$. A subgroup $H$ is open if and only if it is a neighbourhood of the identity $e$. Indeed, if $U \subseteq H$ is a neighbourhood of $e$, then $hU \subseteq H$ is a neighbourhood of each $h \in H$.