Here I want to summarize the various kinds of topological ground-state degeneracy in condensed matter physics and want to know whether there exists any other kind of topological degeneracy. For convenience, let's consider a 2D lattice system with $N$ lattice sites, and we may consider the finite system with open boundary condition(OBC) or periodic boundary condition(PBC). Now there are two kinds of topological degeneracy:
(1)The topological degeneracy is approximate for finite $N$ no matter with OBC or PBC, and it becomes exact degenerate only under the thermodynamic limit($N\rightarrow \infty $). Example: FQHE.
(2)The topological degeneracy(>1) is exact for any finite $N$ with PBC(on a torus), and it's nondegenerate for any finite $N$ with OBC. Example: Kitaev's toric code model.
Is there any other kind(in the above sense) of topological degeneracy?