STUDIES IN LOGIC, GRAMMAR AND RHETORIC 10 (23) 2007
Jordan’s Proof of the Jordan Curve Theorem
Thomas C. Hales
University of Pittsburgh
Abstract.
This article defends Jordan’s original proof of the Jordan curve
theorem.
The celebrated theorem of Jordan states that every simple closed curve in the
plane separates the complement into two connected nonempty sets: an interior
region and an exterior. In 1905, O. Veblen declared that this theorem is “justly
regarded as a most important step in the direction of a perfectly rigorous mathe
matics” [13]. I dedicate this article to A. Trybulec, for moving us much further “in
the direction of a perfectly rigorous mathematics.”
1
Introduction
Critics have been unsparing in their condemnation of Jordan’s original proof. Ac
cording to Courant and Robbins, “The proof given by Jordan was neither short
nor simple, and the surprise was even greater when it turned out that Jordan’s
proof was invalid and that considerable e
ff
ort was necessary to fill the gaps in his
reasoning” [2]. A web page maintained by a topologist calls Jordan’s proof “com
pletely wrong.” Morris Kline writes that “Jordan himself and many distinguished
mathematicians gave incorrect proofs of the theorem. The first rigorous proof is due
to Veblen” [8]. A di
ff
erent Kline remarks that “Jordan’s argument did not su
ffi
ce
even for the case of a polygon” [7].
Dissatisfaction with Jordan’s proof originated early. In 1905, Veblen complained
that Jordan’s proof “is unsatisfactory to many mathematicians. It assumes the
theorem without proof in the important special case of a simple polygon and of the
argument from that point on, one must admit at least that all details are not given”
[13]. Several years later, Osgood credits Jordan with the theorem only under the
assumption of its correctness for polygons, and further warns that Jordan’s proof
contains assumptions.
1
Nearly every modern citation that I have found agrees that the first correct
proof is due to Veblen in 1905 [13]. See, for example, [9, p. 205].
This research has been supported by NSF grant 0503447.
1
“Es sei noch auf die Untersuchungen von C. Jordan verwiesen, wo der Satz, unter
Annahme seiner Richtigkeit f¨ur Polygone, allgemein f¨ur Jordansche Kurven begr¨undet
wird. Jordan beweist hiermit mehr als die Funktionentheorie gebraucht; dagegen macht
er Voraussetzungen, welche diese Theorie streng begr¨undet wissen will” [11].
ISBN 9788374311281
ISSN 0860–150X
45
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Th. C. Hales
My initial purpose in reading Jordan was to locate the error. I had completed a
formal proof of the Jordan curve theorem in January 2005 and wanted to mention
Jordan’s error in the introduction to that paper [3]. In view of the heavy criticism
of Jordan’s proof, I was surprised when I sat down to read his proof to find nothing
objectionable about it. Since then, I have contacted a number of the authors who
have criticized Jordan, and each case the author has admitted to having no direct
knowledge of an error in Jordan’s proof. It seems that there is no one still alive
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 Spring '10
 jim
 Logic, Manifold, Line segment, General topology, Jordan Curve Theorem

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