# Compactness under homeomorphisms

1. Oct 4, 2007

### ehrenfest

1. The problem statement, all variables and given/known data
I want to show that homeomorphism preserve compactness on a set or a space. The definition of a homeomorphism is a continuous function with a continuous inverse.
The definition of a continuous function is a function such that the pre-image of an open set is open.

Let f: X to Y be continuous. Let X be compact
So, if you have an open cover in X then you have a finite subcover. But if you have an open cover in Y, then you could map it to X but how would you know that it is still an open cover in X?

2. Relevant equations

3. The attempt at a solution

2. Oct 4, 2007

### morphism

Suppose f:X->Y is a homeomorphism. If {U_i} is an open cover of Y, then {f^-1(U_i)} is an open cover of X.

3. Oct 4, 2007

### ehrenfest

That is exactly the statement form of my question. Could you please explain why that is true?

4. Oct 4, 2007

### morphism

We know that f is continuous and that each U_i is open. It follows that each f^-1(U_i) is open.

If Y = $\cup$ U_i, can you guess what $\cup$ f^-1(U_i) should be? Can you prove it?

5. Oct 4, 2007

### ehrenfest

Say, that there an x in X s.t. x is not in $\cup$ f^-1(U_i). Then take a nbhd of x N_x. f(N_x) must be in $\cup$ U_i.

OK. So, it comes down to whether the function is injective. Can we assumed it is? That is not part of the definition, is it?

6. Oct 4, 2007

### morphism

A homeomorphism is a bijection, so yes it's injective... But what does that have to do with anything?

Look, Y = $\cup$ U_i. So X = f^-1(Y) = f^-1($\cup$ U_i) = ...

Last edited: Oct 4, 2007
7. Oct 4, 2007

### ehrenfest

OK. I guess the problem is more fundamental. I do not understand why a homeomorphism is a bijection. How would one show from the definition I wrote in the first post that it is surjective? That it is injective?

8. Oct 4, 2007

### morphism

If it has an inverse it must be a bijection.

By the way, the f^-1's I used above were used to indicate taking the preimage.

9. Oct 4, 2007

### ehrenfest

I see. Thanks.