Separation Theorems in the Plane¶
10 Separation Theorems in the Plane
10.1 Jordan Separation Theorem
10.1.1 (Lem 61.1) Let C be a compact subspace of S^{2}; let b be a point of S^{2} – C; and let h be a homeomorphism of S^{2} – b with \mathbb{R}^{2}. Suppose U is a component of S^{2} -C. If U does not contain b, then h(U) is a bounded component of \mathbb{R}^{2} – h(C). If U contains b, then h(U-b) is the unbounded component of \mathbb{R}^{2} – h(C).
10.1.2 (Nulhomotopy lemma) (Lem 61.2) : Let a and b be points of S^{2}. Let A be a compact space, and let f: A->S^{2} – a – b be a continuous map. If a and b lie in the same component of S^{2} – f(A), then f is nulhomotopic.
10.1.3 A separates X ( a subset A of a connected space X ) : X – A is not connected
10.1.4 A separates X into n components ( a subset A of a connected space X ): X-A has n components
10.1.5 Simple closed curve : a space homeomorphic to the unit circle S^{1}.
10.1.6 (Jordan separation theorem) (Thm 61.3) : Let C be a simple closed curve in S^{2}. Then C separates S^{2}.
10.1.6.1 Pf) (Thm 59.1)(Thm 55.3) (Nulhomotopy lemma)
10.1.7 (General Separation theorem) (Thm 61.4) Let A_{1} and A_{2} be closed connected subsets of S^{2} whose intersection consists of precisely two points a and b. Then the set C = A_{1} \cup A_{2} separates S^{2}.
10.2 Invariance of Domain
10.2.1 (Homotopy extension lemma) (Lem 62.1) Let X be a space such that X \times I is normal. Let A be a closed subspace of X, and let f : A->Y be a continuous map, where Y is an open subspace of \mathbb{R}^{n}. If f is nulhomotopic, then f may be extended a continuous map g : X ->Y that is also nulhomotopic.
10.2.1.1 Pf) (Tietze Extension theorem)
10.2.2 (Borsuk lemma) (Lem 62.2) Let a and b be points of S^{2}. Let A be a compact space, and let f : A -> S^{2} – a – b be a continuous injective map. If f is nulhomotopic, then a and b lie in the same component of S^{2} – f(A).
10.2.2.1 Pf) (preceding lemma) Let A be a compact subspace of \mathbb{R} ^{2} - \mathbf{0}. If the inclusion j : A -> \mathbb{R} ^{2} - \mathbf{0} is nulhomotopic, then \mathbf{0} lies in the unbounded component of \mathbb{R} ^{2} – A.
10.2.3 (Invariance of domain) (Thm 62.3) If U is an open subset of \mathbb{R} ^{2} and f: U-> \mathbb{R} ^{2} is continuous and injective, then f(U) is open in \mathbb{R} ^{2} and the inverse function f^{-1} : f(U) -> U is continuous.
10.2.3.1 Pf) (Borsuk lemma)
10.3 Jordan Curve theorem
10.3.1 (Thm 63.1) Let X be the union of two open sets U abd V, such that U \cap V can be written as the union of two disjoint open sets A and B. Assume that there is a path \alpha in U from a point a of A to a point b of B, and that there is a path \beta in V from b to a. Let f be the loop f = \alpha * \beta.
10.3.1.1 The path-homotopy class [ f ] generates an infinite cyclic subgroup of \pi_{1} (X,a).
10.3.1.2 If \pi_{1} (X,a) is itself infinite cyclic, it is generated by [ f].
10.3.1.3 Assume there is a path \gamma in U from a to the point a’ of A, and that there is a path \delta in V from a’ to a. Let g be the loop g = \gamma * \delta. Then the subgroups of \pi_{1} (X,a) generated by [ f ] and [ g ] intersect in the identity element alone.
10.3.2 (Nonseparation theorem) (Thm 63.2) Let D be an arc in S^{2}. Then D does not separate S^{2}.
10.3.2.1 Pf) (Borsuk lemma) (Thm 63.1)
10.3.3 (General Nonseparation theorem) (Thm 63.3) Let D_{1} and D_{2} be closed subsets of S^{2} s.t. S^{2} – D_{1} \cap D_{2} is simply connected. If neither D_{1} nor D_{2} separates S^{2}, then D_{1} \cup D_{2} does not separate S^{2}.
10.3.4 (Jordan Curve theorem) (Thm 63.4) Let C be a simple closed curve in S^{2}. Then C separates S^{2} into precisely two components W_{1} and W_{2}. Each of the sets W_{1} and W_{2} has C as its boundary; that is C = \bar{W_{i}} – W_{i} for i = 1,2.
10.3.4.1 Pf) (Jordan separation theorem) (thm 63.1)
10.3.5 (Thm 63.5) Let C_{1} and C_{2} be closed connected subsets of S^{2} whose intersection consists of two points. If neither C_{1} nor C_{2} separates S^{2}, then C_{1} \cup C_{2} separates S^{2} into precisely two componenets.
10.3.6 (Schoenflies theorem) : If C is a simple closed curve in S^{2} and U and V are the components of S^{2} – C, then \bar{U} and \bar{V} are each homeomorphic to the closed unit ball B^{2}.
10.4 Imbedding graphs in the plane
10.4.1 Linear graph G : a Hausdorff space that is written as the union of finitely many arcs.
10.4.1.1 Complete graph on n vertices : G contains exactly n vertices, and if for every pair of distinct vertices of G there is an edge of G joining them
10.4.2 Theta space : a Hausdorff space that is written as the union of three arcs A, B, and C, each pair of which intersects precisely in their end points.
10.4.3 (Lem 64.1) Let X be a theta space that is a subspace of S^{2}; let A, B and C be the arcs whose union is X. Then X separates S^{2} into three components, whose boundaries are A\cup B , B \cup C, and A \cup C, respectively. The component having A \cup B as its boundary equals one of the components of S^{2} – A \cup B.
10.4.3.1 Pf) (Thm 63.5)
10.4.4 (Thm 64.2) Let X be the utilities graph. Then X cannot be imbedded in the plane.
10.4.4.1 Utilities graph : given three houses h_{1}, h_{2}, h_{3}, and three utilities, g (gas), w(water), and e(electricity), Can you connect each utility to each house without letting any of the connecting lines cross?
10.4.5 (Lem 64.3) Let X be a subspace of S^{2} that is a complete graph on four vertices a_{1}, a_{2} , a_{3} and a_{4}. Then X separates S^{2} into four components. The boundaries of these components are the sets X_{1}, X_{2}, X_{3} and X_{4}, where X_{i} is the union of those edges of X that do not have a_{i} as a vertex.
10.4.6 (Thm 64.4) The complete graph on five vertices cannot be imbedded in the plane.
10.5 Winding number of a simple closed curve
10.5.1 (Lem 65.1) Let G be a subspace of S^{2} that is a complete graph on four vertices a_{1}, …, a_{4}. Let C be the subgraph a_{1}a_{2}a_{3}a_{4}a_{1}, which is a simple closed curve. Let p and q be interior points of the edges a_{1}a_{3} and a_{2}a_{4}, respectively. Then
10.5.1.1 The points p and q lie in different components of S^{2} – C.
10.5.1.2 The inclusion j : C -> S^{2} – p – q induces an isomorphism of fundamental groups.
10.5.1.2.1 Pf) (Lem 64.1)(Lem 64.3)
10.5.1.3 Let C be a simple closed curve in S^{2}; let p and q lie in different components of S^{2} – C. Then the inclusion mapping j : C -> S^{2} – p – q induces an isomorphism of fundamental groups.
10.5.1.3.1 Pf) (Cor 58.5)
10.6 Cauchy integral formula
10.6.1 Winding number of f with respect to a ( a point a not in the image of a loop f in \mathbb{R}^{2}) : Let g(s) = [f(s) – a] / \Vert f(s) – a \Vert then g is a loop in S^{1}. Let p : \mathbb{R} -> S^{1} be the standard covering map, and let \bar{g} be a lifting of g to S^{1}. Because g is a loop, the difference \bar{g}(1) - \bar{g}(0) is an integer. This integer is winding number of f w.r.t. a .
10.6.1.1 n( f, a ) <- winding number of f with respect to a
10.6.2 Free homotopy F between the loops f_{0} and f_{1} (a continuous map F : I \times I -> X ) : Let F(0,t) = F(1,t) for all t. Then for each t, the map f_{t}(s) = F(s,t) is a loop in X.
10.6.2.1 It is a homotopy of loops in which the base point of the loop is allowed to move during the homotopy.
10.6.3 (Lem 66.1) Let f be a loop in \mathbb{R}^{2} – a.
10.6.3.1 If \bar{f} is the reverse of f, then n(\bar{f} , a) = -n (f, a).
10.6.3.2 If f is freely homotopic to f’, through loops lying in \mathbb{R}^{2}-a, then n(f,a) = n(f’, a).
10.6.3.3 If a and b lie in the same component of \mathbb{R}^{2} – f(1), then n(f,a) = n(f,b).
10.6.3.4 Simple loop f (a loop f in X) : f(s) = f(s’) only if s = s’ or if one of the points s, s’ is 0 and the other is 1.
10.6.3.4.1 If f is a simple loop, its image set is a simple closed curve in X.
10.6.3.5 (Thm 66.2) Let f be a simple loop in \mathbb{R}^{2}. If a lies in the unbounded component of \mathbb{R}^{2} – f(1), then n(f,a) = 0; while if a lies in the bounded component, n(f,a) = \mp 1.
10.6.3.5.1 Pf) (Thm 54.5)
10.6.3.6 Counterclockwise loop f (a simple loop f in \mathbb{R}^{2}) : n(f,a) = +1 for some a (and hence for every a) in the bounded componenet of \mathbb{R}^{2} – f(1).
10.6.3.6.1 Clockwise loop f if n(f,a) = -1.
10.6.3.7 (Lem 66.3) Let f be a piecewise-differentiable loop in the complex plane; let a be a point not in the image of f. Then n(f,a) = \frac{1}{2 \pi i} \int_{f} \frac{dz}{z-a}
10.6.3.7.1 Definition of the winding number of f : \frac{1}{2 \pi i} \int_{f} \frac{dz}{z-a}
10.6.3.8 (Cauchy integral formula-classical version) (Thm 66.4) Let C be a simple closed piecewise-differentiable curve in the complex plane. Let B be the bounded component of \mathbb{R}^{2} – C. If F(z) is analytic in an open set \Omega that contains B and C, then for each point a of B, F(a) = \mp \frac{1}{2 \pi i} \int_{C} \frac{F(z)}{z-a} dz
10.6.3.8.1 + sign if C is oriented counterclockwise, and – otherwise.