Connectedness and topological sectors
Disconnected regions of a configuration space cannot be connected by an allowed continuous path.
$$\pi_0(X)=\{\text{path components of }X\}$$
What this means
Disconnected regions of a configuration space cannot be connected by an allowed continuous path.
Deep dive: mathematical and physical mechanism
Path-connected components classify the zeroth homotopy invariant. In quantum phases, the relevant space may be the set of gapped Hamiltonians respecting selected symmetries. Two sectors are distinct if every allowed interpolation encounters a gap closing or violates a protecting symmetry.
How it appears in quantum phenomena
Distinct phases of a one-dimensional chiral chain can occupy different gapped sectors.
Assumptions, caveats and limitations
The classification depends on the allowed deformations and symmetries.
Connect the ideas
For context, see the Foundations of Topology topic guide and the reading references.