Denjoy–Riesz theorem

In topology, the Denjoy–Riesz theorem describes a class of sets of points in the Euclidean plane that can be covered by a continuous image of the unit interval, without self-intersections (a Jordan arc). 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.

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. As Moore & Kline (1919) describe, 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.

By applying this theorem to a two-dimensional version of the Smith–Volterra–Cantor set, it is possible to find an Osgood curve, a Jordan arc or closed Jordan curve whose Lebesgue measure is positive.

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