Topology in mathematics studies the properties of an object that are preserved under continuous deformations.
For example, because an unknot (a) cannot be continuously transformed into a trefoil knot (b) without cutting it (not a continuous deformation) they are not topologically equivalent.
For example, because an unknot (a) cannot be continuously transformed into a trefoil knot (b) without cutting it they are not topologically equivalent.
The object that is studied in condensed matter physics, is the Hamiltonian.
Two Hamiltonians are topologically equivalent whenever an Hamiltonian can be continuously transformed into another Hamiltonian.
Unlike a knot that can be visualized in space, the topology of the quantum Hall state manifests itself in momentum space.