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Take a residually finite group $G$ and a subgroup $H$ such that no finite number of conjugates of $H$ intersect at 1trivially, but all conjugates intersect by 1have trivial intersection. Now declare that subgroup and all finite index subgroups of $G$ open. $G$ becomes a locally compact non-discrete group and $H$ is open and does not contain normal non-trivial subgroups.
Take a residually finite group $G$ and a subgroup $H$ such that no finite number of conjugates of $H$ intersect at 1, but all conjugates intersect by 1. Now declare that subgroup and all finite index subgroups of $G$ open. It is $G$ becomes a locally compact non-discrete group and $H$ is open and does not contain normal non-trivial subgroups.
Take a residually finite group $G$ and a subgroup $H$ such that no finite number of conjugates of $H$ intersect at 1, but all conjugates intersect by 1. Now declare that subgroup and all finite index subgroups open. It is a locally compact non-discrete group and $H$ is open and does not contain normal non-trivial subgroups.