Is not a Polish group

Abstract

The group $\text{PL}_{+}(I)$ of increasing piecewise-linear self-homeomorphisms of the interval $I=[0,1]$ may not be assigned a topology in such a way that it becomes a Polish group. The same statement holds for the groups $\text{Homeo}_{+}^{\text{Lip}}(I)$ of bi-Lipschitz homeomorphisms of $I$, and $\text{Diff}_{+}^{1+\unicode[STIX]{x1D716}}(I)$ of diffeomorphisms of $I$ whose derivatives are Hölder continuous with exponent $\unicode[STIX]{x1D716}$, as well as the corresponding groups acting on the real line and on the circle.