Exercise 12 Chapter 7

Gary

I have made a slight change to my solution to this exercise.

While working on exercise 15 I noticed that in the introduction to it on page 373 Needham says that he doesn't want to rely on the TAP for the proof of Brouwer's FP theorem, because he doesn't want to limit the proof to the case where $g$ is a continuous mapping having a finite number of $p$-points in a finite region.

This caused me to look back at exercise 12 and I noticed that in the proof of Rouché's theorem in part (ii) I had used the TAP in the final step. However in part (iii) of 12 we are only asked to deduce that $\nu[(f+g)(\Gamma),0]=\nu[f(\Gamma),0]$, and not take the final step and claim this as a proof of Rouché's theorem. In my answer however I had in fact made this final step and I now think that Needham was thinking about exercise 15 and so in exercise 12 he only asks us to take part (iii) so far, which enables him to use it in the proof of the first part of exercise 15. In this way the result of exercise 15 (Brouwer's FP theorem) does not rely on the TAP, which is just what Needham wants.

So I have altered my answer to 12 part (iii) so that it points out that if we want to go a step further and prove Rouché's theorem we must accept the restrictions on $g$ that the TAP brings with it.

Do you think this is right?

Vasco