I am studying the KT-transition, which is called a topological phase transition. The phase transition is driven by vortices in a 2-D superfluid, where it is shown that at a critical temperature $T_c$ free vortices are energetically favoured over vortex-antivortex pairs.
There is no spontaneous symmetry breaking in this phase transition. Instead it is called a topological phase transition. However, in my very naive understanding of topolopy (obtained from looking at donuts and pretzels and stuff), we are interested in the structure, or the singularities of the system. Even in the case of the transition where these dislocations/vortices/singularities become free, the number of singularities doesn't necessarily change. I would say that the topology is conserved, since only the distance between the singularities is changed.
If the topology of the system doesn't change, why do we call it a topological phase transition?