The Kosterlitz-Thouless transition is often described as a "topological phase transition." I understand why it isn't a conventional Landau-symmetry-breaking phase transition: there is no local symmetry-breaking order parameter on either side of the transition, and all thermodynamic quantities remain continuous (though not analytic) at all derivative orders across the transition. I also understand why it is "topological" in the sense that it involves the binding of vortex topological defects.
But when I think of a "topological phase transition," I think of a phase transition between two phases characterized by different topological quantum numbers, which are topological invariants over the entire phase. For example, the transition between a trivial and topological band insulator corresponds to a change in the Chern number of the band structure (between zero and nonzero). The transition between a topologically trivial and topologically ordered phase corresponds to a change in the topologically protected ground-state degeneracy (between 1 and greater than 1).
Is the KT transition, in which neither side of the transition has either long-range entanglement or symmetry-protected topological order, a topological phase transition in this sense? If so, what are the quantum numbers on both sides of the transition? (I'm not counting trivial answers like "the topological quantum number defined as 0 if vortices are bound and 1 if they are unbound.")