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 in the plane separates it into precisely two components, of which is the common boundary.
Lemma 1 Let be a compact subspace of , let be a point of ; and let be a homeomorphism of with . Suppose is a component of . If does not contain , then is a bounded component of . If contains , then is the bounded component of .
In particular, if has components, then has components.
Proof We show first that if is a component of , then is connencted.
Let be the set of components of ; let . Because is locally connected, the set are connected, disjoint open subsets of . Therefore the sets are connected, disjoint, open subsets of , so the sets are the components of .
Lemma 2 (Nulhomotopy lemma) Let and be points of . Let be a compact space, and let
be a continuous map. If and lie in the same component of , then is nulhomotopic.
Definition 1 If is a connected space and , we say that separates if is not conneted; if has components, we say that separates into components.
Definition 2 An arc is a space homeomorphic to the unit inverval . The end points of are the two points and such that and are connected; the other points of are called interior points of .
A simple closed curve is a space homeomorphic to the unit circle .
Theorem 1Suppose , where and are open sets of . Suppose that is path connected, and that . Let and be the inclusion mappings of and , respectively, into . Then the images of the induced homomorphisms
Theorem 2 (The Jordan separation theorem) Let be a simple closed curve in . Then separate .
2 Invariance of domain
Lemma 3 (Homotopy extension lemma) Let be a space such that is normal. Let be a closed subspace of , and let be a continuous map, where is an open subspace of . If is nulhomotopic, then may be extended to a continuous map that is also nulhomotoptic.
The following lemma is partial converse to the nulhomotopy lemma of the preceding section.
Lemma 4 (Borsuk lemma) Let and be points of . Let be a compact space, and let be a continuous injective map. If is nulhomotopic, then and lie in the same component of .
Theorem 3 (Invariance of domain) If is an open subset o and is continuous and injective, then is open in and the inverse function 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 in the case where the intersection is path connected. In the next theorem, we examine what happens when 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 be a simple closed curve in . Then separates into precisely two components and . Each of the sets and has as its boundary; that is, for .