Proving Invariance of Domain Theorem

In summary, the conversation was about proving the Invariance of domain theorem for spring break homework. Bredon's book "Topology and Geometry" and Hatcher's book were recommended as sources for the proof, with the latter also mentioning the use of singular homology. Wiki also mentions a proof involving Brouwer's fixed point theorem, which can be found in Milnor's book "Topology from the differentiable viewpoint". The conversation ends with a link to a forum discussing the topic.
  • #1
Ja4Coltrane
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0
Hello,

For spring break homework, I'm supposed to prove the Invariance of domain theorem (stating that continuous injective functions from an open set in R^n to R^n are open maps).

Does anyone know of any books/sources of any kind which will help?

Thanks!
 
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  • #2
It is 19.8 in Bredon's book "Topology and Geometry". It appears as a corollary of the generalized Jordan curve theorem. For the proof of the generalized Jordan curve theorem however, I recommend the book of Hatcher (http://www.math.cornell.edu/~hatcher/AT/ATch2.pdf). It is Proposition 2B.1 (b) there.

The arguments in these proofs are not difficult to understand but they do rely very much on the theory of singular homology. If you are not familiar with the theory of homology, Wiki talks about a proof involving Brouwer's fixed point theorem (http://en.wikipedia.org/wiki/Invariance_of_domain). Brouwer's fixed point theorem can be proved with only elementary concepts of differential topology (See Milnor's book Topology from the differentiable viewpoint) so you would probably prefer that route, but I do not know of a book where that proof of Invariance of domain can be found. Please let me know if you find such a book!

Googling I found the following: http://at.yorku.ca/cgi-bin/bbqa?for...sk=show_msg;msg=1480.0001.0001.0001.0001.0001
 

1. What is the Invariance of Domain Theorem?

The Invariance of Domain Theorem is a fundamental theorem in topology that states that a continuous bijection (a function that is both one-to-one and onto) between two Euclidean spaces of the same dimension is a homeomorphism (a continuous function with a continuous inverse). In simpler terms, it means that a continuous mapping of one space onto another will preserve the topological properties of the original space.

2. Why is the Invariance of Domain Theorem important?

The Invariance of Domain Theorem is important because it allows mathematicians to use the tools and techniques of topology to study and understand other mathematical objects and structures. It also has many applications in other fields, such as physics and engineering, where understanding and manipulating topological properties is crucial.

3. How is the Invariance of Domain Theorem proven?

The Invariance of Domain Theorem can be proven using techniques from algebraic topology, specifically the concept of homotopy. The proof involves showing that a continuous bijection between two Euclidean spaces of the same dimension can be deformed into a homeomorphism while preserving the topological properties of the original space.

4. Can the Invariance of Domain Theorem be generalized to higher dimensions?

Yes, the Invariance of Domain Theorem can be generalized to higher dimensions. The theorem holds for any two Euclidean spaces of the same dimension, regardless of how high that dimension may be.

5. Are there any exceptions to the Invariance of Domain Theorem?

Yes, there are a few exceptions to the Invariance of Domain Theorem. One exception is when the two spaces in question have different dimensions. Another exception is when the continuous mapping involves a space with a boundary. In these cases, the theorem does not hold and different techniques must be used to prove similar results.

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