Proving Invariance of Domain Theorem

[Ja4Coltrane]

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!

 

[quasar987]

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