1780 CHAPTER 56. APPROXIMATION BY RATIONAL FUNCTIONS

while for all z ∈ Bm

∑k=1

n(Γk,z) = 0.

Definition 56.2.5 Let γ be a closed curve in an open set, Ω,γ : [a,b]→Ω. Then γ is said tobe homotopic to a point, p in Ω if there exists a continuous function, H : [0,1]× [a,b]→Ω

such that H (0, t) = p,H (α,a) = H (α,b) , and H (1, t) = γ (t) . This function, H is calleda homotopy.

Lemma 56.2.6 Suppose γ is a closed continuous bounded variation curve in an open set,Ω which is homotopic to a point. Then if a /∈Ω, it follows n(a,γ) = 0.

Proof: Let H be the homotopy described above. The problem with this is that it is notknown that H (α, ·) is of bounded variation. There is no reason it should be. Therefore,it might not make sense to take the integral which defines the winding number. There arevarious ways around this. Extend H as follows. H (α, t) = H (α,a) for t < a,H (α, t) =H (α,b) for t > b. Let ε > 0.

Hε (α, t)≡ 12ε

∫ t+ 2ε

(b−a) (t−a)

−2ε+t+ 2ε

(b−a) (t−a)H (α,s)ds, Hε (0, t) = p.

Thus Hε (α, ·) is a closed curve which has bounded variation and when α = 1, this con-verges to γ uniformly on [a,b]. Therefore, for ε small enough, n(a,Hε (1, ·)) = n(a,γ)because they are both integers and as ε → 0,n(a,Hε (1, ·))→ n(a,γ) . Also, Hε (α, t)→H (α, t) uniformly on [0,1]× [a,b] because of uniform continuity of H. Therefore, for smallenough ε, you can also assume Hε (α, t) ∈ Ω for all α, t. Now α → n(a,Hε (α, ·)) is con-tinuous. Hence it must be constant because the winding number is integer valued. But

limα→0

12πi

∫Hε (α,·)

1z−a

dz = 0

because the length of Hε (α, ·) converges to 0 and the integrand is bounded because a /∈Ω.Therefore, the constant can only equal 0. This proves the lemma.

Now it is time for the great and glorious theorem on simply connected regions. Thefollowing equivalence of properties is taken from Rudin [113]. There is a slightly differentlist in Conway [32] and a shorter list in Ash [7].

Theorem 56.2.7 The following are equivalent for an open set, Ω ̸= C.

1. Ω is homeomorphic to the unit disk, B(0,1) .

2. Every closed curve contained in Ω is homotopic to a point in Ω.

3. If z /∈Ω, and if γ is a closed bounded variation continuous curve in Ω, then n(γ,z) =0.

4. Ω is simply connected, (Ĉ\Ω is connected and Ω is connected. )