## non-trivial topology

Hi there, I've come across the term 'non-trivial topology' or 'non-trivial surface states' when researching topological superconductors and really need a bit of help as to exactley what this means? I've tried google but no-one seems to give a definition?
Many thanks for checking this out
 Wouldn't it just be a topology that isn't trivial? http://en.wikipedia.org/wiki/Trivial_topology
 Yea I was just hoping someone could explain it a bit nicer!

## non-trivial topology

Oh ok then, I'll try to do better. I do have to warn you that I'm not particularly knowledgeable in topology or it's application to superconductors.

In the definition of a topology, one specifies a set, and a collection of subsets. Roughly speaking, the specified subsets tell you the abstracted geometry of the set. So for example, in the real number line you have the familiar definition of open set as the specification of the subsets. These open set can always be made smaller, and this is an important feature of the real numbers. Then consider the discrete topology, where every single point set is an open set. In this case you cannot always make an open set smaller. This would be a good choice of topology for the integers, for example.

So again, roughly speaking the topology carries some information on the "shape" of the set. The trivial topology, on the other hand, can be imposed on any set. So clearly, the trivial topology fails to tell you this kind of information. If this isn't clear, I'll make another example. If you try to put the same topology of the real numbers on the integers, you'll end up with the discrete topology( (-a,a) will eventually only contain 0 as you make a smaller). However, you could easily put the trivial topology on both sets.

So non trivial topologies are topologies that have enough structure to tell you something about the set.

I'm not sure what a "non trivial surface state" is. Hopefully someone will come along who knows more about that than I.