On the Cantor set and the Cantor-Lebesgue functions

TL;DR

This paper reviews properties of the Cantor set, the Cantor-Lebesgue functions, and space-filling curves, emphasizing their construction, extensions, and relation to Hausdorff's theorem, serving as a systematic exposition rather than original research.

Contribution

It provides a comprehensive review of classical properties and constructions related to the Cantor set and associated functions, including space-filling curves, with no new theoretical innovations.

Findings

Detailed properties of the Cantor set and its ternary expansion

Construction and properties of the Cantor-Lebesgue function and its extension

Discussion of Lebesgue's space-filling curves and Hausdorff's theorem

Abstract

The ternary Cantor set $\mathcal{C}$, constructed by George Cantor in 1883, is the best known example of a perfect nowhere-dense set in the real line. The present article we study the basic properties $\mathcal{C}$ and also study in detail the ternary expansion characterization $\mathcal{C}$. We then consider the Cantor-Lebesgue function defined on $\mathcal{C},$ prove its basic properties and study its continuous extension to $[0,1].$ We also consider the geometric construction of $F$ as the uniform limit of polygonal functions. Finally, we consider the Lebesgue's function defined from $\mathcal{C}$ onto $[0,1]^2$ and onto $[0,1]^3,$ as well as their continuous extension to $[0,1],$ i.e., obtained the Lebesgue's space filling curves. Finally we discuss Hausdorff's theorem, which is a natural generalization of the definition of Lebesgue's functions, that states that any compact metric space is a continuous image of the Cantor set $\mathcal{C}$. This notes are the outgrowth of a (zoom)-seminar during the 2020 spring and summer semesters based on H. Sagan's book \cite{sagan}, thus there is not any pretension of originality and/or innovation in this notes and its main objective is a systematic exposition of these topics, in other words its objective is being a review.