Where do you need topology in physics?

[Gerenuk]

I admit I hardly know anything about topology, but I have the feeling it is a heavy, abstract part of mathematics. Yet, I heard that it can be important for physics.

In which areas concepts of topology are crucial so that results cannot be guessed by common sense alone? Where are these results essential to the physicist and not merely a prove of existence of a feature?

 

[confinement]

General point-set topology is not really that useful, because nobody has proposed any use for non-Hausdorf or other more general spaces, but point-set topology is a pre-requisite for Algebraic and Differential topology, and modern differential geometry, which have many applications in theoretical physics.

The reason that point set topology is not useful is that all of the spaces in physics are at least topological manifolds (Hausdorf space which is locally homeomorphic to R^n) or even differentiable manifolds (locally diffeomorphic to R^n) and/or metric spaces (all diff manifolds are metrizable, and so are the Hilbert spaces in QM).

One of the reasons that mathematicians began to study manifolds is because they naturally arise from differential equations. In classical mechanics, Hamilton's equations especially can be treated with differential forms on manifolds, and this allows a deeper understanding of intregrable systems (see KAM-Theorem). In chaos theory differential topology is used constantly e.g. the relation between attractor reconstruction and the embedding theorem.

Any modern understanding of GR, particle theory, or many fields of condensed matter theory, depends on understanding algebraic and differential topology.