What is so topological about topological phase transitions?

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?

• My feeling is that the most general meaning of "topological phase transition" is "cannot be described by spontaneous breaking of a local symmetry". – Norbert Schuch Jan 17 at 10:31

The importance of the topological defects in phase transitions have been emphasized by Kosterlitz and Thouless, which shared 2016 Nobel prize for theoretical discoveries of topological phase transitions and topological phases of matter''.