Jordan-Brouwer Separation Theorem and Invariance of Domain

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Discussion Overview

The discussion revolves around the relationship between the Jordan-Brouwer Separation Theorem and the Invariance of Domain, exploring how one theorem may imply the other and the role of injectivity in their proofs. The scope includes theoretical aspects and mathematical reasoning.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the relationship between the Jordan-Brouwer Separation Theorem and the Invariance of Domain, noting their understanding of the concepts but seeking clarity on their connection.
  • Another participant questions the claim of equivalence between the two theorems, suggesting that both are long-standing statements that were difficult to prove until the advent of homology.
  • A participant proposes that zeroth homology can measure components of a space, which may be used to demonstrate the separation theorem, linking it to the Invariance of Domain by discussing the properties of open sets in Euclidean space.
  • Another participant suggests a weak direction of implication from Invariance of Domain to the separation theorem, discussing the implications of connected complements and open sets.
  • One participant acknowledges the clarity provided by a previous response, indicating that the explanation made sense to them.
  • A participant notes that the weak direction of implication is uncertain, suggesting that it may not be true that an injection from the boundary of a ball extends to an injection on the ball itself.
  • Another participant introduces Tietze's theorem, speculating on its applicability to extending continuous maps from closed subspaces to the whole space.
  • There is a question raised about the necessity of injectivity in the context of these theorems.
  • One participant expresses uncertainty about extending an injection into another injection, prompting a reference to the Alexander horned sphere as a counterexample.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the relationship between the Jordan-Brouwer Separation Theorem and the Invariance of Domain, with multiple competing views and uncertainties expressed throughout the discussion.

Contextual Notes

Some claims depend on assumptions about the properties of continuous maps and the nature of injectivity, which remain unresolved. The discussion also highlights the complexity of extending injections and the implications of connectedness in the context of these theorems.

Anonymous217
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So I'm having trouble understanding how these two are related, i.e., how one proves the other.

I understand the ideas behind both of them: For J-B, you're basically taking R^n and throwing in a sphere, so the inside of the sphere is bounded and everything outside the sphere is unbounded. For Invariance of Domain, it's pretty obvious just by the definition (the image of an open subset of R^n is open).
However, I don't really see a relationship between the two. Can anyone give some insight?

Also, I was curious why we only need injectivity for both of them, where surjectivity is basically unnecessary in any possible proof. What makes 1-1 important?
 
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I was unaware that these theorems were equivalent. Where did you hear such a claim?

The only way I know them to be related is that they are both obvious-looking statement that no one has been able to prove for a long time. That is, until homology came along.
 
(zeroth) homology measures components of a space hence can be used to prove the separation thm. components of an open set in Euclidean space are open, so knowing about components can be used to prove inv of domain.

If S is an n-1 sphere and B an n ball, and f an injective continuous map, separation implies the complement of the image of f(S) in R^n has 2 components and the complement of f(B) has only one.

since f(B-S) is connected, it is thus a component of the open complement of f(S), hence f(B-S) is open.

Since every open set is a union of such balls B, this shows that Jordan separation implies invariance of domain.
 
as a weak version of the other direction, suppose f is an injective continuous map from B to R^n, and we assume f(B) has connected complement. Then if we knew that f(B-S) were open, it would follow, since f(B-S) is also connected, that the complement of f(S) has 2 components. I.e. invariance of domain does imply that the complement of f(S) has one more component than the complement of f(B).
 
^^ This is the type of answer I was looking for. It makes complete sense now; thanks.
 
notice the weak direction is weak because it is not clear, indeed probably not true, that an injection from S^n-1 to R^n extends to an injection on B^n.
 
you are welcome.
 
Mathwonk:

I think we may be able to use Tietze's theorem that a continuous, real-valued map from a closed subspace of a normal space (I think the ball B^n, as a metric space, is normal, and its boundary S^(n-1) is closed in B^n--the interior seems to be open) X, extends into the whole space, tho I am not sure if there is a version for maps into R^n; maybe we can argue component-wise to get a continuous map.
 
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injectivity?
 
  • #10
I am not sure we can extend an injection into an injection; let me see.
 
  • #11
i believe the alexander horned sphere is a famous counterexample.
 

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