## Nice characterization of convergence

Suppose is a topological space, and consider the sequence with the following property:

- every subsequence has a subsequence converging to .

Then .

## 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.

## Weakly Compact means bounded

Let be a Banach space, and be a compact set in the weak topology. Then is bounded.

## Euclidean spaces of different dimensions are not homeomorphic

I would like to give a proof for the fact that if then and are not homeomorphic.

1) for the case where one of the numbers is , we can give a simple proof

2) for the general proof, we can use the Generalized Jordan Curve Theorem.

(cf. Spivak, Intro to Differential Geometry, Chapter 1, Ex. 8,9)

## Generalized Jordan Curve Theorem

If is homeomorphic to , then has two components, and is the boundary of each.

*Generalized Jordan Curve Theorem*

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