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

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