Cantor Set

The Cantor set ($T_\infty$) is given by taking the interval [0,1] (set $T_0$), removing the middle third ($T_1$), removing the middle third of each of the two remaining pieces ($T_2$), and continuing this procedure ad infinitum. It is therefore the set of points in the Interval [0,1] whose ternary expansions do not contain 1, illustrated below.

This produces the Set of Real Numbers $\{x\}$ such that

$$x={c_1\over 3}+\ldots+{c_n\over 3^n}+\ldots,$$ (1)

where $c_n$ may equal 0 or 2 for each $n$. This is an infinite, Perfect Set. The total length of the Line Segments in the $n$th iteration is

$$\ell_n = \left({2 \over 3}\right)^n,$$ (2)

and the number of Line Segments is $N_n = 2^n$, so the length of each element is

$$\epsilon_n \equiv {\ell \over N} = \left({1\over 3}\right)^n$$ (3)

and the Capacity Dimension is

$$d_{\rm cap} \equiv - \lim_{\epsilon\to 0^+} {\ln N \over\ln\epsilon} = - \lim_{n\to\infty} {n\ln 2\over -n\ln 3}$$

$$= {\ln 2\over\ln 3} = 0.630929\ldots.$$ (4)

The Cantor set is nowhere Dense, so it has Lebesgue Measure 0.

A general Cantor set is a Closed Set consisting entirely of Boundary Points. Such sets are Uncountable and may have 0 or Positive Lebesgue Measure. The Cantor set is the only totally disconnected, perfect, Compact Metric Space up to a Homeomorphism (Willard 1970).

See also Alexander's Horned Sphere, Antoine's Necklace, Cantor Function

References

Boas, R. P. Jr. A Primer of Real Functions. Washington, DC: Amer. Math. Soc., 1996.

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 15-20, 1991.

Willard, S. §30.4 in General Topology. Reading, MA: Addison-Wesley, 1970.