Exercise 3, § 4

Let Γ be a simply transitive group of homeomorphisms of R endowed with the topology of pointwise convergence. For each x ∈ R let s_x denote the element of Γ such that s_x(0) = x. Show that the mapping x → s_x is a homeomorphism of R onto Γ [use the fact that if s ∈ Γ is such that x < s(x) for some x ∈ R, then y < s(y) for all y ∈ R]; hence show that Γ is isomorphic to R as a topological group [use Exercise 17 a) of § 3 and Theorem 1 of Chapter V, § 3].