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Bulletin of the London Mathematical Society
Article . 1980 . Peer-reviewed
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A Proof of the Jordan Curve Theorem

A proof of the Jordan curve theorem
Authors: Tverberg, Helge;

A Proof of the Jordan Curve Theorem

Abstract

Let F be a Jordan curve in the plane, i.e. the image of the unit circle C = {(x,y);x + y = 1} under an injective continuous mapping y into R. The Jordan curve theorem [1] says that / ? 2 \ F is disconnected and consists of two components. (We shall use the original definition whereby two points are in the same component if and only if they can be joined by a continuous path (image of [0,1]).) Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it. The present paper is intended to provide a reasonably short and selfcontained proof or at least, failing that, to point at the need for one.

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Keywords

general Jordan curve, Topology of the Euclidean \(2\)-space, \(2\)-manifolds, Jordan polygon, Topological spaces of dimension \(\leq 1\); curves, dendrites

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
24
Top 10%
Top 10%
Average
bronze
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