## Compact & Hausdorff spaces

The following problem wants to prove that compact and Hausdorff spaces have an interesting property. If is Hausdorff and compact, and we consider another Hausdorff topology coarser than and Hausdorff, then .

1) Prove that a compact set in a Hausdorff space is closed.

2) Every continuous function from a compact space onto a Hausdorff space is open.

3) If you have a bijective continuous function from a compact space to a Hausdorff space, then the two spaces are homeomorphic.

A short application, which was my motivation for this post is the following problem:

Let be a Banach space and be a compact subset in the strong topology. Let be a sequence in such that in the weak topology. Prove that strongly.

**Proof: **1) Suppose is a compact set. We can assume that . Pick . For every there exists an open set which does not intersect an open set . Doing this for every with we get an open cover of , which has a finite subcover . Then for the open set not containing , we can find an open set containing which has nothing in common with this union. The union contains , and therefore is contained in an open set outside . Doing this for every we get the desired result.

2) Any closed set is compact, which is mapped by into a compact set, and by 1) this is closed. Taking complements we see that the image of any open set is open. (at this point we have used that is onto!)

3) is invertible and use 2).

To prove the first statement that any coarser Hausdorff topology is the same as the initial topology, we proceed as follows:

Take , where is compact and Hausdorff and is also Hausdorff, and is the identity mapping. This is a continuous mapping since . It is also bijective, and therefore a homeomorphism. Therefore the two spaces have the same topological structure and .

The intuitive idea is that adding open sets preserves ; removing open sets preserves compactness and if you want to keep both you must leave them as they are. 🙂

The short problem at the end is proven already in the above steps. The weak topology is coarser than the strong topology and Hausdorff, and is a compact space. Therefore the topologies are the same.