The diffeomorphism groups of the real line are pairwise bihomeomorphic

Abstract

For an $r=0,1,\dots ,\infty$, by $D^r\left(R\right)$, $D_{+}^r\left(R\right)$, $D_c^r\left(R\right)$ we denote respectively the groups of $C^r$ diffeomorphisms, orientation-preserving $C^r$ diffeomorphisms, and compactly supported $C^r$ diffeomorphisms of the real line. We think of these groups as bitopologies spaces endowed with the compact-open $C^r$ topology and the Whitney $C^r$ topology. We prove that all the triples $(D^r\left(R\right),D_{+}^r\left(R\right),D_c^r\left(R\right))$, $0\le r\le \infty$, are pairwise bitopologically equivalent, which allows us to apply known results on the topological structure of homeomorphism groups of the real line to recognizing the topological structure of the diffeomorphism groups of $R$.