There exist a large number of, sometimes surprising, applications of Baire category theorem. For instance, in this blog, we saw that
- two generic orientation-preserving homeomorphisms of the circle generate a non abelian free subgroup,
- a generic continuous function is nowhere differentiable.
See the previous notes Free groups acting on the circle and Almost all continuous functions are nowhere differentiable. Now, we want to prove:
Theorem: If , the space is simply connected.
Fix a base point and an enumeration . Given two closed curves satisfying , we want to prove that and are homotopic (relatively to ) in . Let
It is a closed subset of the Banach space of the continuous functions with respect to the supremum norm , so that , endowed with the distance induced by , is a complete metric space. Now, let
Because , the space is simply connected, so that . In fact, it is not difficult to convince ourself that is a dense open subset of . A possible argument is the following:
First, if and if denotes the distance between and (which is positive since is compact), then the ball in of radius and centered at is clearly included into . Therefore, is open in .
Then, let be a bump function whose support is included into and satisfying . Now, given a homotopy , define the perturbation
Of course, and . Furthermore, if is small enough and well chosen, then . This proves that is dense in .
Now, it is sufficient to apply Baire category theorem to conclude that the intersection is non empty. But any element of this intersection defines a homotopy between and satisfying , that is to say a homotopy in .
Therefore, we have proved that any two closed curves in based at are homotopic in . This shows that the group is trivial, ie., the space is simply connected.