Cantor set
Historical remarks

Cantor introduced what we call today the Cantor ternary set C {\displaystyle {\mathcal {C}}} as an example "of a perfect point-set, which is not everywhere-dense in any interval, however small." Cantor described C {\displaystyle {\mathcal {C}}} in terms of ternary expansions, as "the set of all real numbers given by the formula: z = c 1 / 3 + c 2 / 3 2 + ⋯ + c ν / 3 ν + ⋯ {\displaystyle z=c_{1}/3+c_{2}/3^{2}+\cdots +c_{\nu }/3^{\nu }+\cdots } where the coefficients c ν {\displaystyle c_{\nu }} arbitrarily take the two values 0 and 2, and the series can consist of a finite number or an infinite number of elements."

A topological space P {\displaystyle P} is perfect if all its points are limit points or, equivalently, if it coincides with its derived set P ′ {\displaystyle P'}. Subsets of the real line, like C {\displaystyle {\mathcal {C}}}, can be seen as topological spaces under the induced subspace topology.

Cantor was led to the study of derived sets by his results on uniqueness of trigonometric series. The latter did much to set him on the course for developing an abstract, general theory of infinite sets.

Benoit Mandelbrot wrote much on Cantor dusts and their relation to natural fractals and statistical physics. He further reflected on the puzzling or even upsetting nature of such structures to those in the mathematics and physics community. In The Fractal geometry of Nature, he described how "When I started on this topic in 1962, everyone was agreeing that Cantor dusts are at least as monstrous as the Koch and Peano curves," and added that "every self-respecting physicist was automatically turned off by a mention of Cantor, ready to run a mile from anyone claiming C {\displaystyle {\mathcal {C}}} to be interesting in science."
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