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. $G$ 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 trivially, but all conjugates have 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 open. It is 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. $G$ becomes a locally compact non-discrete group and $H$ is open and does not contain normal non-trivial subgroups.