Posts Tagged ‘topology’

Nice characterization of convergence

October 6, 2011 Leave a comment

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.

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Categories: Topology Tags: , , ,

Compact & Hausdorff spaces

March 1, 2011 Leave a comment

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.

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Weakly Compact means bounded

February 28, 2011 Leave a comment

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

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Euclidean spaces of different dimensions are not homeomorphic

November 27, 2010 3 comments

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

November 27, 2010 Leave a comment

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.

Generalized Jordan Curve Theorem
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Categories: Analysis, Topology Tags: ,
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