Denjoy–Riesz theorem
{{Short description|A compact set of totally disconnected points in the plane can be covered by a Jordan arc}}
File:Julia set (Rev formula 02).jpg. By the Denjoy–Riesz theorem, there exists an arc passing through all the points in this set.]]
In topology, the Denjoy–Riesz theorem states that every compact set of totally disconnected points in the Euclidean plane can be covered by a continuous image of the unit interval, without self-intersections (a Jordan arc).
Definitions and statement
A topological space is zero-dimensional according to the Lebesgue covering dimension if every finite open cover has a refinement that is also an open cover by disjoint sets. A topological space is totally disconnected if it has no nontrivial connected subsets; for points in the plane, being totally disconnected is equivalent to being zero-dimensional. The Denjoy–Riesz theorem states that every compact totally disconnected subset of the plane is a subset of a Jordan arc.{{citation
| last = Krupka | first = Demeter
| doi = 10.2991/978-94-6239-073-7
| isbn = 978-94-6239-072-0
| mr = 3290001
| page = 158
| publisher = Atlantis Press, Paris
| series = Atlantis Studies in Variational Geometry
| title = Introduction to global variational geometry
| url = https://books.google.com/books?id=Wf6bclM5UGwC&pg=PA158
| volume = 1
| year = 2015}}.
History
{{harvtxt|Kuratowski|1968}} credits the result to publications by Frigyes Riesz in 1906, and Arnaud Denjoy in 1910, both in Comptes rendus de l'Académie des sciences.{{citation
| last = Kuratowski | first = K. | authorlink = Kazimierz Kuratowski
| mr = 0259835
| page = 539
| publisher = Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw
| series = New edition, revised and augmented. Translated from the French by A. Kirkor
| title = Topology. Vol. II
| url = https://books.google.com/books?id=GbrSBQAAQBAJ&pg=PA539
| year = 1968| isbn = 9781483271798 }}. As {{harvtxt|Moore|Kline|1919}} describe,{{citation
| last1 = Moore | first1 = R. L. | author1-link = Robert Lee Moore
| last2 = Kline | first2 = J. R. | author2-link = John Robert Kline
| doi = 10.2307/1967872
| issue = 3
| journal = Annals of Mathematics
| mr = 1502556
| pages = 218–223
| series = Second Series
| title = On the most general plane closed point-set through which it is possible to pass a simple continuous arc
| volume = 20
| year = 1919| jstor = 1967872 }}. Riesz actually gave an incorrect argument that every totally disconnected set in the plane is a subset of a Jordan arc. This generalized a previous result of L. Zoretti, which used a more general class of sets than Jordan arcs, but Zoretti found a flaw in Riesz's proof: it incorrectly presumed that one-dimensional projections of totally disconnected sets remained totally disconnected. Then, Denjoy (citing neither Zoretti nor Riesz) claimed a proof of Riesz's theorem, with little detail. Moore and Kline state and prove a generalization that completely characterizes the subsets of the plane that can be subsets of Jordan arcs, and that includes the Denjoy–Riesz theorem as a special case.
References
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