A topological group G is a group, which is also a topological space, such that multiplication G×G → G given by (g₁,g₂) ↦ g₁g₂ and inversion G → G given by g ↦ g^–1 are both continuous functions. Examples include ℝ, ℚ, the circle S¹, and the groups listed under matrix groups. A topological group which is a (topological) manifold is necessarily a Lie group.