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

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 $\mathop{\rm Diff}_+^{1+ε}(I)$ of diffeomorphisms of $I$ whose derivatives are Hölder continuous with exponent $ε$.

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