Topological type of the group of uniform homeomorphisms of the real line

AbstractIn this paper, we study the group Hu(R) of uniform homeomorphisms having the uniform topology together with two subgroups H∞(R)={h∈Hu(R);lim|x|→∞(h(x)−x)=0} and Hc(R), the group of compact support homeomorphisms. We show that the group Hu(R) is homeomorphic to ℓ∞×2ℵ0 and that the triple (Hu(R)0,H∞(R),Hc(R)) is homeomorphic to (ℓ∞×ℓ2×ℓ2,{0}×ℓ2×ℓ2,{0}×ℓ2×ℓ2f), where 2ℵ0 denotes a discrete space of the cardinality of the continuum, while Hu(R)0 is the identity component of Hu(R) and ℓ2f is the linear hull of the standard orthonormal basis of the separable Hilbert space ℓ2. We will also discuss the relations among Hu(R)0 and some function spaces containing it