### Archive

Posts Tagged ‘topology’

## Nice characterization of convergence

Suppose $X$ is a topological space, and consider the sequence $(x_n)$ with the following property:

• every subsequence $(x_{n_k})$ has a subsequence converging to $x$.

Then $x_n \to x$.

Categories: Topology Tags: , , ,

## Compact & Hausdorff spaces

The following problem wants to prove that compact and Hausdorff spaces have an interesting property. If $(X,\mathcal{T})$ is Hausdorff and compact, and we consider another Hausdorff topology $\mathcal{T}^\prime$ coarser than $\mathcal{T}$ and Hausdorff, then $\mathcal{T}=\mathcal{T}^\prime$.

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 $E$ be a Banach space and $K\subset E$ be a compact subset in the strong topology. Let $(x_n)$ be a sequence in $K$ such that $x_n\to x$ in the weak topology. Prove that $x_n \to x$ strongly.

## Weakly Compact means bounded

Let $E$ be a Banach space, and $A \subset E$ be a compact set in the weak topology. Then $A$ is bounded.

## Euclidean spaces of different dimensions are not homeomorphic

I would like to give a proof for the fact that if $m\neq n,\ m,n \in \Bbb{N}^*$ then $\Bbb{R}^n$ and $\Bbb{R}^m$ are not homeomorphic.

1) for the case where one of the numbers $m,n$ is $1$, 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 $A \subset \Bbb{R}^n$ is homeomorphic to $S^{n-1}$, then $\Bbb{R}^n \setminus A$ has two components, and $A$ is the boundary of each.