  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.
