Are the complements of homeomorphic compact connected subsets homeomorphic?

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In summary, the conversation discusses the homeomorphism of the complements of two compact connected subsets in the plane. It is believed that if A and B are homeomorphic, then their complements, C and D, have an equal number of bounded components that are contractible and/or simply connected. The theorem is true for open, simply connected subsets of the plane, but may not hold for pathological examples in 2D. The main question is how to prove that if A and B are homeomorphic, then their complements also have connected components that are simply connected.
  • #1
AKG
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Let A and B be compact connected subsets of the plane, homeomorphic to one another. Are their complements homeomorphic?

I think they are. If A and B are empty, there's nothing to prove. Otherwise, let C denote the complement of A, let D denote the complement of B. We look at the connected components of C and D. Each will have one unbounded component with one hole in it. The unbounded component of C should be homeomorphic to the unbounded component of D. The rest of the components of C are bounded and, I believe, contractible and/or simply connected (are the two equivalent in the plane?). I want to argue that since A and B are homeomorphic, C and D have an equal number of these bounded components. I also want to argue that any two bounded, open, connected, contractible and/or simply connected subsets of the plane are homeomorphic. Actually, it might even be that any two open contractible subsets of the plane are homeomorphic. If this is so, then in technical terms, I believe we can find a bijection f from the collection of connected components of C to the collection of connected components of D such that X is homeomorphic to f(X). Since each X is open, I would like to argue that this induces a homeomorphism from all of C to all of D.

Is the above good?
 
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  • #2
Is this a homework question, or something you want to use to prove a homework question?
 
  • #3
Neither...
 
  • #4
Then why is it in the homework section?
 
  • #5
Oops, force of habit I guess. It should be moved, but it doesn't matter to me.
 
  • #6
I believe this theorem is true, although it does not generalize to R3, with alexander's horned sphere as a counterexample. Your argument isn't quite rigorous, and it will take some work for it to handle pathological examples similar to the horned sphere that may arise in 2D.
 
  • #7
Any two open, simply connected subsets of the plane are homeomorphic. We know that thanks to the Riemann mapping theorem in complex analysis.
 
  • #8
It seems the hard part to prove is the following:

Suppose A and B are connected compacts in the plane, homeomorphic to one another. If AC has a connected component that is simply connected, so does BC.

It seems obvious, but how would one go about proving it?
 

1. What is a homeomorphic complement?

A homeomorphic complement is a mathematical concept that refers to a set of elements that are necessary to complete a given set, in such a way that the resulting set is homeomorphic to the original set. In simpler terms, it is a set of elements that, when added to a given set, do not change its topological structure.

2. How is a homeomorphic complement different from a topological complement?

A homeomorphic complement and a topological complement are different in that the former is a set of elements that maintains the topological structure of the original set, while the latter is a set of elements that completely reverses the topological structure of the original set.

3. What are some examples of homeomorphic complements?

Some examples of homeomorphic complements include a single point added to a circle, a single line added to a sphere, or a single point removed from a plane. These examples maintain the topological structure of the original set while completing it.

4. How are homeomorphic complements useful in mathematics?

Homeomorphic complements are useful in mathematics for a variety of reasons. They can help to simplify complex topological structures, aid in the study of abstract spaces, and provide a way to compare different topological spaces.

5. Can homeomorphic complements be applied to real-world situations?

Yes, homeomorphic complements can be applied to real-world situations, particularly in fields such as computer science, engineering, and physics. They can be used to model and analyze complex systems, such as networks or physical structures, and provide insights into their topological properties.

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