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Jordan curve theorem
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(Theorem)
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Informally, the Jordan curve theorem states that every Jordan curve divides the Euclidean plane into an ``outside'' and an ``inside''. The proof of this geometrically plausible result requires surprisingly heavy machinery from topology. The difficulty lies in the great generality of the statement and inherent difficulty in formalizing the exact meaning of words like ``curve'', ``inside'', and ``outside.''
There are several equivalent formulations.
Theorem 2 If $\Gamma$ is a simple closed curve in the sphere $S^2$ then $S^2 \setminus \Gamma$ consists of precisely two connected components.
Theorem 3 Let $h: \RR \to \RR^2$ be a one-to-one continuous map such that $|h(t)| \to \infty$ as $|t| \to \infty$ Then $\RR^2 \setminus h(\RR)$ consists of precisely two connected components.
The two connected components mentioned in each formulation are, of course, the inside and the outside the Jordan curve, although only in the first formulation is there a clear way to say what is out and what is in. There we can define ``inside'' to be the bounded connected component, as any picture can easily convey.
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"Jordan curve theorem" is owned by rmilson. [ full author list (4) | owner history (1) ]
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| Keywords: |
Jordan curve, simple closed curve |
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Cross-references: bounded, clear, continuous map, one-to-one, sphere, simple closed curve, topology, proof, Euclidean plane, Jordan curve
There are 4 references to this entry.
This is version 6 of Jordan curve theorem, born on 2002-11-11, modified 2004-02-16.
Object id is 3587, canonical name is JordanCurveTheorem.
Accessed 8417 times total.
Classification:
| AMS MSC: | 54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces ) | | | 54A05 (General topology :: Generalities :: Topological spaces and generalizations ) |
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Pending Errata and Addenda
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