We now present an example of a group with an uncountable number of elements that is not a Lie group under a certain topology. The group given by with a fixed irrational number, is a subgroup of the torus that is not a Lie group when given the subspace topology. If we take any small neighborhood of a point in , for example, the portion of in is disconnected. The group winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a dense subgroup of .
We now present an example of a group with an uncountable num?ber of elements that is not a Lie group under a certain topology. The group given by with a fixed irrational number, is a subgroup: of the torus that is not a Lie group when given the subspace topology. If we take any small neighborhood of a point in , for example, the portion of in is disconnected. The group winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a dense subgroup of .