372 CHAPTER 14. FUNDAMENTALS

22. Suppose Ω is an open connected subset of C with the property that if Γ is a simpleclosed curve contained in Ω and UΓ, VΓ respectively the bounded and unboundedcomponents of ΓC, then UΓ ⊆Ω. Show that Ω is simply connected. Hint: If Ĉ\Ω =

A∪B where A,B separate Ĉ \Ω with ∞ ∈ B, argue that B \ {∞} is closed and A iscompact. Show dist(A,B\{∞}) = δ > 0. Then make a grating of finitely manyhorizontal and vertical lines equally spaced with distance between successive linesno more than δ/10 such that the small squares cover up A. Now consider the verticeswhich are contained in Ω. Each has four lines or maybe two lines emanating fromit. Start at such a vertex p and travel over sides of squares, never going back theway you came along one of these sides till you return to the point p at which youstarted. Delete all paths of the form p̂, p̂1, ..., p̂ where p̂ ̸= p. The result must be asimple closed curve Γ contained in Ω,∞ /∈UΓ. By assumption UΓ does not containany points of A. Since you could have started at any point of this grating, and thereare finitely many vertices, A must be empty. A more systematic way of doing thisis in the material on cycles in next chapter showing that the points of A must beincluded.