with a ∈ ∈ R ∖ ∖ Q a fixed irrational number, is a subgroup of the torus T 2 that is not a Lie group when given the subspace topology. If we take any small neighborhood U of a point h in H , for example, the portion of H in U is disconnected. The group H winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a dense subgroup of T 2 .