804 CHAPTER 23. DEGREE THEORY

for each x ∈U . Explain why the above function of x is measurable. Now by Eggo-roff’s theorem, there is measurable set A of measure less than ε

mn10n such that off A,the convergence is uniform. Let Ck be a countable union of non overlapping halfopen rectangles one of which is of the form ∏

ni=1(ai,bi] such that each has diameter

less than 2−k. Consider the half open rectangles which have nonempty intersectionwith Sm \A,Ik. Then repeat the argument given in the first section of this chapter.Show that for k large enough, the rank condition and uniform convergence aboveimplies that mn (∪{f(I) : I ∈Ik}) is less than ε . Now show that f(A) is contained ina set of measure no more than mn10n ε

mn10n = 2ε . Thus f(Sm) has measure no morethan 3ε . Since ε is arbitrary, this establishes the desired conclusion.

20. Let X be a Banach space and let Ω be a symmetric and bounded open set. LetF : Ω→ X be odd and compact 0 /∈ (I−F)(∂Ω). Show using Corollary 23.9.3 thatD(I−F,Ω,0) is an odd integer.

21. Let F be compact. Suppose I − F is one to one on B(0,r). Then using similarreasoning to the finite dimensional case, show that there is a δ > 0 such that

(I−F)(0)+B(0,δ )⊆ (I−F)(B(0,r))

22. Let F be compact. Suppose I−F is locally one to one on an open set Ω. Show that(I−F) maps open sets to open sets. This is a version of invariance of domain.

23. Suppose (I−F) is locally one to one and F is compact. Suppose also that

lim∥x∥→∞

∥(I−F)x∥= ∞.

Show that (I−F) is onto.

24. As a variation of the above problem, suppose F : X → X is compact and

lim∥x∥→∞

∥F (x)∥∥x∥

= 0

Then I−F is onto. Note that I−F is not one to one.

25. Suppose F is compact and ∥(I−F)x− (I−F)y∥≥ α ∥x− y∥. Show (I−F) is onto.

26. The Jordan curve theorem is: Let C denote the unit circle,{(x,y) ∈ R2 : x2 + y2 = 1

}.

Suppose γ : C→ Γ ⊆ R2 is one to one onto and continuous. Then R2 \Γ consistsof two components, a bounded component (called the inside) U1 and an unboundedcomponent (called the outside), U2. Also the boundary of each of these two compo-nents of R2 \Γ is Γ and Γ has empty interior. Using the Jordan separation theorem,prove this important result.