PL_+(I) is not a Polish group. (6th October 2015) - COHEN, MICHAEL P.; KALLMAN, ROBERT R.
Abstract : The group 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 Homeo_+^Lip(I) of bi-Lipschitz homeomorphisms of I, and Diff_+^1+[STIX]x1D716(I) of diffeomorphisms of I whose derivatives are Hölder continuous with exponent [STIX]x1D716, as well as the corresponding groups acting on the real line and on the circle.