In 1966, Mermin and Wagner proved that there can be no spontaneous breaking of a continuous symmetry at non-zero temperature in spatial dimensions greater than or equal to two. This might be taken to mean that for non-zero temperature the order is short-range, i.e. that correlation functions decay exponentially with distance. However, in seminal works a few years later, Berezinskii, Kosterlitz and Thouless (BKT) noted that for the marginal case, d=2, there is an intriguing third option: quasi-long-range-order, in which the spatial correlation functions exhibit power-law decay. The finite-temperature transition from the quasi-long-range-ordered to the short-range-ordered state is described by BKT as a proliferation of vortices, and is a prime example of the importance of topologically non-trivial excitations in condensed matter physics. In this talk, I shall examine how the BKT theory must be modified when there is more than one candidate for a spontaneously broken symmetry: for example, in the case of a supersolid, where both the O(2) symmetry corresponding to the superfluid phase and the O(2) symmetry corresponding to the translation of the solid lattice are broken simultaneously. The picture that emerges will be of the vortices gradually becoming "non-topological" as a particular symmetric point of the theory is approached - their cores are, as it were, "eaten away" by spin waves of the higher-symmetry theory. This leads to a suppression of the BKT temperature to zero at the high-symmetry point, giving a phase diagram that looks like - but crucially is not - that of a quantum critical system..