$\mathop{\rm PL}_+(I)$ is not a Polish group

Abstract

The group $\mathop{\rm PL}_+(I)$ 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 $\mathop{\rm Homeo}_+^{Lip}(I)$ of bi-Lipschitz homeomorphisms of $I$, and $\mathop{\rm Diff}_+^{1+\epsilon}(I)$ of diffeomorphisms of $I$ whose derivatives are H\"older continuous with exponent $\epsilon$.