Denjoy–Riesz theorem

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).

[1] 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.

[2] As Moore & Kline (1919) 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.

[4] A related result is the analyst's traveling salesman theorem, describing the point sets that form subsets of curves of finite arc length.