Lebesgue’s Space-Filling Curve

In a paper on infinite linear point manifolds written in 1883, in which Cantor searched for a characterization of the continuum, he offers in the appendix the set of all points that can be represented by $$frac{{2{t_1}}}{3} + \frac{{2{t_2}}}{{{3^2}}} + \frac{{2{t_3}}}{{{3^3}}} + \frac{{2{t_4}}}{{{3^4}}} + ...,$$ where t _j = 0 or 1, as an example of a perfect set (a set that is equal to the set of all its accumulation points) that is not dense in any interval, no matter how small (Cantor [2]). This set, which had its humble beginnings as a counterexample in an appendix has since taken on a life of its own and has served ever since its inception, as an example, counterexample, and inspiration for inquiries into the most remote recesses of mathematical analysis. It is no coincidence that it appears in a fundamental role in the study of space-filling curves. We will see in Chapter 6 that this role is even more fundamental than it would appear to be from a study of the present chapter.