Separation theorems in the plane

1 The Jordan separation theorem

Our proof of the Jordan curve theorem divides into three parts. The first, which we call the Jordan separation theorem, states that a simple closed curve in the plane separates it into at least two components. The second says that an arc in the plane does not separate the plane. And the third, the Jordan curve theorem proper, says that a simple closed curve C in the plane separates it into precisely two components, of which C is the common boundary.

Lemma 1 Let C be a compact subspace of S^2, let b be a point of S^2\setminus C; and let h be a homeomorphism of S^2\setminus b with \mathbf{R}^2. Suppose U is a component of S^2\setminus C. If U does not contain b, then h(U) is a bounded component of \mathbf{R}^2\setminus h(C). If U contains b, then h(U\setminus b) is the bounded component of \mathbf{R}^2\setminus h(C).

In particular, if S^2\setminus C has n components, then \mathbf{R}^2\setminus h(C) has n components.

Proof  We show first that if U is a component of S^2\setminus C, then U\setminus \{b\} is connencted.

Let (U_\alpha) be the set of components of S^2\setminus C; let V_\alpha=h(U_\alpha\setminus\{b\}). Because S^2\setminus C is locally connected, the set U_\alpha are connected, disjoint open subsets of S^2. Therefore the sets V_\alpha are connected, disjoint, open subsets of \mathbf{R}^2\setminus h(C), so the sets V_\alpha are the components of \mathbf{R}^2\setminus h(C).

Lemma 2 (Nulhomotopy lemma) Let a and b be points of S^2. Let A be a compact space, and let

\displaystyle f:A\to S^2\setminus\{a,b\}

be a continuous map. If a and b lie in the same component of S^2\setminus f(A), then f is nulhomotopic.

Definition 1 If X is a connected space and A\subset X, we say that A separates X if X\setminus A is not conneted; if X\setminus A has n components, we say that A separates X into n components.

Definition 2 An arc is a space homeomorphic to the unit inverval [0,1]. The end points of A are the two points p and q such that A\setminus\{p\} and A\setminus\{q\} are connected; the other points of A are called interior points of A.

simple closed curve is a space homeomorphic to the unit circle S^1.

Theorem 1Suppose X=U\cup V, where U and V are open sets of X. Suppose that U\cap V is path connected, and that x_0\in U\cap V. Let i and j be the inclusion mappings of U and V, respectively, into X. Then the images of the induced homomorphisms

\displaystyle i_*:\pi_1(U,x_0)\to \pi_1(X,x_0)\ \ \text{and} \ \ j_*:\pi_1(V,x_0)\to \pi_1(X,x_0)

generate \pi_1(X,x_0).

Theorem 2 (The Jordan separation theorem) Let C be a simple closed curve in S^2. Then C separate S^2.

2 Invariance of domain

Lemma 3 (Homotopy extension lemma) Let X be a space such that X\times I is normal. Let A be a closed subspace of X, and let f:A\to Y be a continuous map, where Y is an open subspace of \mathbf{R}^n. If f is nulhomotopic, then f may be extended to a continuous map g:X\to Y that is also nulhomotoptic.

The following lemma is partial converse to the nulhomotopy lemma of the preceding section.

Lemma 4 (Borsuk lemma) Let a and b be points of S^2. Let A be a compact space, and let f:A\to S^2\setminus\{a,b\} be a continuous injective map. If f is nulhomotopic, then a and b lie in the same component of S^2\setminus f(A).

Theorem 3 (Invariance of domain) If U is an open subset o \mathbf{R}^2 and f:U\to\mathbf{R}^2 is continuous and injective, then f(U) is open in \mathbf{R}^2 and the inverse function f^{-1}:f(U)\to U is continuous.

3 The Jordan curve theorem

The special case of the Seifert-van Kampen theorem that we used in proving the Jordan separation theorem tell us something about the fundamental group of the space X=U\cup V in the case where the intersection U\cap V is path connected. In the next theorem, we examine what happens when U\cap V is not path connected. This result will enable us to complete the proof the Jordan curve theorem.

Now we prove the Jordan curve theorem

Theorem (The Jordan curve theorem) 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=\overline{W}_i-W_i for i=1,2.

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