218 CHAPTER 8. POSITIVE LINEAR FUNCTIONALS

Then if B is a curve from x to y, it must, by assumption, intersect C and so it mustintersect Ĵ. Let a be the first point of intersection of Ĵ and b its last. Then the curvexaby where ab goes along Ĵ avoids C and so C does not separate Ui after all. Explainall this.

30. Let J be a simple closed curve in the plane with the interior component Ui. Let z bea point on J and let x be some point of Ui. Show there exists a simple curve joiningz and x. Hint: Fill in the details. Let xn → z where xn ∈Ui the xk being distinctpoints, k = 0,1,2,3... and x0 = x. Let an denote a strictly increasing sequence ofpositive numbers increasing to 1 with a0 = 0. Then let γn : [an−1,an]→Ui such thatγn (an−1) = xn−1,γn (an) = xn and γ∗n∩ γ∗k = /0 if |n− k|> 1 while γ∗n∩ γ∗n−1 = xn−1.Let γ (t) ≡ γn (t) for t ∈ [an−1,an] and γ (1) ≡ z. You could let these γn be squarecurves and use the result of the above problem.