Compactness under homeomorphisms

  • Thread starter ehrenfest
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In summary, a homeomorphism is a continuous function with a continuous inverse. It is defined as a bijection between two spaces. The property of preserving compactness means that if {U_i} is an open cover of Y, then {f^-1(U_i)} is an open cover of X. This is true because a homeomorphism is a bijection, so it is both injective and surjective.
  • #1
ehrenfest
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Homework Statement


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?



Homework Equations





The Attempt at a Solution

 
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  • #2
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
morphism said:
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.

That is exactly the statement form of my question. Could you please explain why that is true?
 
  • #4
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 = [itex]\cup[/itex] U_i, can you guess what [itex]\cup[/itex] f^-1(U_i) should be? Can you prove it?
 
  • #5
Say, that there an x in X s.t. x is not in [itex]\cup[/itex] f^-1(U_i). Then take a nbhd of x N_x. f(N_x) must be in [itex]\cup[/itex] 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
A homeomorphism is a bijection, so yes it's injective... But what does that have to do with anything?

Look, Y = [itex]\cup[/itex] U_i. So X = f^-1(Y) = f^-1([itex]\cup[/itex] U_i) = ...
 
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  • #7
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
If it has an inverse it must be a bijection. :smile:

By the way, the f^-1's I used above were used to indicate taking the preimage.
 
  • #9
I see. Thanks.
 

FAQ: Compactness under homeomorphisms

What is compactness under homeomorphisms?

In mathematics, compactness under homeomorphisms refers to a property of topological spaces where the image of a compact set under a homeomorphism is also compact. In simpler terms, it means that if a set is compact in one topological space, it will also be compact in another topological space that is homeomorphic to the first one.

Why is compactness under homeomorphisms important?

Compactness under homeomorphisms is important because it allows us to study the properties of topological spaces by considering a smaller set of spaces that are homeomorphic. It also helps in proving theorems and making generalizations about topological spaces.

What is the relationship between compactness and homeomorphisms?

The relationship between compactness and homeomorphisms is that a homeomorphism preserves compactness. This means that if a set is compact in one topological space, it will also be compact in another topological space that is homeomorphic to the first one.

How is compactness under homeomorphisms different from compactness under continuous functions?

Compactness under homeomorphisms is a stronger condition than compactness under continuous functions. While a homeomorphism is a bijective and continuous function, a continuous function does not necessarily have to be bijective. This means that compactness under homeomorphisms is a more restrictive condition than compactness under continuous functions.

What are some examples of topological spaces that exhibit compactness under homeomorphisms?

One example is the unit interval [0,1] in the standard topology, which is homeomorphic to any other interval [a,b] where a and b are real numbers. Another example is the unit circle S^1 in the standard topology, which is homeomorphic to any other circle with a different radius. In general, any two topological spaces that are homeomorphic to each other will exhibit compactness under homeomorphisms.

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