In the context of physics, topological phases of matter often reveal themselves by causing certain quantities that are usually continuous to change in discrete

In the context of physics, topological phases of matter often reveal themselves by causing certain quantities that are usually continuous to change in discrete steps instead. A key example is the electrical conductivity of certain quantum Hall effect systems, which jumps in discrete steps (is "quantized" in physics jargon) as an ever stronger magnetic field is applied to the system in a transverse direction. The connection to the donut/bagel thing more familiar in topology is that topology also deals with integers: a manifold can have 0,1,2,3... holes in it, that sort of thing. One key mathematical quantity that comes up in the works of Thouless and Haldane is the Chern number https://en.wikipedia.org/wiki/Chern_class https://en.wikipedia.org/wiki/Chern_class The reason the physics quantities jump is that the Chern numbers of certain electron bands change, and by definition a Chern number is an integer.