The Mermin-Wagner theorem states that continuous symmetry cannot be spontaneously broken at any finite temperature in two dimensions or lower.
The Kosterlitz-Thouless (KT) transition is a phase transition on a symmetric system (no easy axis for mangetic moments to align) in two dimensions. I believe it can be said that the Kosterlitz-Thouless system has continuous symmetry, please correct me if I am wrong.
At zero temperature the KT system is a ferromagnetic state, which means all the magnetic moments are pointing in the same direction (some random direction since no easy axis). Since all of the magnetic moments are pointing in the same direction (chosen at random) at zero temperature then we can say that the continuous symmetry of the KT system has spontaneously broken. However this does not violate the Mermin-Wagner theorem since this occurs at zero temperature.
At small finite temperatures the KT system is no longer a ferromagnetic state. Instead vortex anti-vortex pairs form. Doesn't this mean that the continuous symmetry of the KT system is broken again at non-zero finite temperatures. Thus, violating the Mermin-Wagner theorem?