Topology – a Global Perspective
Topological Coffee Set: What Is What Here?
Topology is concerned with those properties of geometrical objects that remain unchanged when the object is deformed. The exact shape, on the other hand, cannot be seen with the means of topology. With the coffee set, the number of holes is decisive: It is a topological property. Thus, from the point of view of topology, a donut resembles a coffee cup. If you look at a coffee pot, you might first think that it has three holes: one on the handle, one where the coffee is poured into the pot, and one for pouring. In fact, the last two together are only one hole. You can picture this by deforming the coffee pot in your mind to such an extent that it becomes an object with two holes.
... of matter are the domain of Topological Physics. It studies, among other things, the distinct electrical behavior at the edge and inside of a material. With the quantum Hall effect, the quantization of the Hall conductivity results from a global topological invariant – an integer number comparable to the number of holes in a coffee cup or donut. Even flaws or impurities in the material used do not change the physical surface properties, hence the Nobel-Prize-worthy high precision of the effect!