The group H can, however, be given a different topology, in which the distance between two points h 1 , h 2 ∈ ∈ H is defined as the length of the shortest path in the group H joining h 1 to h 2 . In this topology, H is identified homeomorphically with the real line by identifying each element with the number θ θ in the definition of H . With this topology, H is just the group of real numbers under addition and is therefore a Lie group.