Версия от 10:42, 26 февраля 2024; EducationBot(обсуждение | вклад)(Новая страница: «{{Английская Википедия/Панель перехода}} {{Short description|A compact set of totally disconnected points in the plane can be covered by a Jordan arc}} thumb|upright=1.2|A totally disconnected [[Julia set. 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 T...»)
(разн.) ← Предыдущая версия | Текущая версия (разн.) | Следующая версия → (разн.)
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).
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.[1]
History
Шаблон:Harvtxt credits the result to publications by Frigyes Riesz in 1906, and Arnaud Denjoy in 1910, both in Comptes rendus de l'Académie des sciences.[2] As Шаблон:Harvtxt describe,[3] 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.[3]
A related result is the analyst's traveling salesman theorem, describing the point sets that form subsets of curves of finite arc length. 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.
.jpg&action=edit&redlink=1){kind=link}
.jpg){kind=link}