topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, 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 In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysi ...

, 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 In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\su ...

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.

History

credits the result to publications by

Frigyes Riesz Frigyes Riesz (, , sometimes known in English and French as Frederic Riesz; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> ...

in 1906, and Arnaud Denjoy in 1910, both in ''

Comptes rendus de l'Académie des sciences (, ''Proceedings of the Academy of Sciences''), or simply ''Comptes rendus'', is a French scientific journal published since 1835. It is the proceedings of the French Academy of Sciences. It is currently split into seven sections, published o ...

''. As 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.

Applications and related results

By applying this theorem to a two-dimensional version of the

Smith–Volterra–Cantor set In mathematics, the Smith–Volterra–Cantor set (SVC), ε-Cantor set, or fat Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–V ...

, it is possible to find an

Osgood curve In mathematical analysis, an Osgood curve is a non-self-intersecting curve that has positive area. Despite its area, it is not possible for such a curve to cover any Domain (mathematical analysis), two-dimensional region, distinguishing them from ...

, a Jordan arc or closed Jordan curve whose

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...

is positive.. For an earlier construction of a positive-area Jordan curve, not using this theorem, see . A related result is the analyst's traveling salesman theorem, describing the point sets that form subsets of curves of finite

arc length Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...

. Not every compact totally disconnected set has this property, because some compact totally disconnected sets require any arc that covers them to have infinite length.

References

{{DEFAULTSORT:Denjoy-Riesz Theorem General topology Theorems in topology