Section 3.1 Another look at the Cantor set
Cantor constructed his set in the 1880's to help him understand a problem in Fourier series. While the set seemed unnatural to mathematicians of the time, it has become a central example in real analysis. Cantor's construction is as follows. Start with the unit interval \(I = [0,1]\text{,}\) the set of all real numbers between 0 and 1 inclusive. Remove the open middle third \(\left(\frac{1}{3},\frac{2}{3}\right)\) of the interval \(I\) to obtain the two intervals \(I_1 = \left[0,\frac{1}{3}\right]\) and \(I_2 = \left[\frac{2}{3},1\right].\) Then remove the open middle thirds of the intervals \(I_1\) and \(I_2\) to obtain the intervals \(I_{1,1} = \left[0,\frac{1}{9}\right],\) \(I_{1,2} = \left[\frac{2}{9},\frac{1}{3}\right],\) \(I_{2,1} = \left[\frac{2}{3},\frac{7}{9}\right],\) and \(I_{2,2} = \left[\frac{8}{9},1\right].\) Repeating this process inductively, we obtain \(2^n\) intervals of length \(1\left/3^n\right.\) at the \(n^{\text{th}}\) stage. The cantor set \(C\) consists of all those points in \(I\) which are never removed at any stage. More precisely, if \(C_n\) denotes the union of all of the intervals left after the \(n^{\text{th}}\) stage of the construction, then
This process is illustrated in Figure 3.1
It's clear that \(C\) should be self-similar, since the effect of the construction on the intervals \(I_1\) and \(I_2\) is the same as the effect on the whole interval \(I\text{,}\) but on a smaller scale. Thus \(C\) consists of two copies of itself scaled by the factor \(1/3\text{.}\)
The Cantor set has many non-intuitive properties. In some sense, it seems very small; if we were to assign a ``length'' to it, that length would have to be zero. Indeed, by it's very construction it is contained in \(2^n\) intervals of length \(1/3^n\text{.}\) Thus the length of \(C_n\) is \(2^n/3^n\) which tends to zero as \(n\rightarrow \infty .\) Since \(C\) is contained in \(C_n\) for all \(n\text{,}\) the length of \(C\) must be zero. It might even appear that there is nothing left in \(C\) after tossing so much out of the original interval \(I.\) In reality, the Cantor set is a very rich set with infinitely many points. Recall that only open intervals are removed during the construction. Thus all of the infinitely many endpoints remain. For example, \(1/3\text{,}\) \(2/3\text{,}\) and \(80/81\) are all in \(C.\) There are still many more points in \(C,\) however.
There is a general technique for finding points of the Cantor set. The first stage in the construction consists of the two intervals \(I_1\) and \(I_2.\) Choose one and discard the other. Now the interval we chose, say \(I_1\) for concreteness, contains two disjoint intervals, \(I_{1,1}\) and \(I_{1,2},\) in the next stage of the construction. Choose one of those and discard the other. If we continue this process inductively, we obtain a nested sequence of closed intervals which will collapse down to a point in the Cantor set. For example, we might have chosen the interval \(I_1\) at the first stage. Then we could have chosen the interval \(I_{1,2}\) at the next stage. We might then choose to alternate between the first or second sub-interval at any point generating intervals of the form \(I_{1,2,1,2,\text{...}1,2}.\) These intervals collapse down to a single point which is not the endpoint of any removed interval.
The process for finding points in \(C\) constructs a one to one correspondence with the set of infinite sequences of 1s and 2s. The sequence corresponding to a particular point in \(C\) might be called the address of that point. As we will see, this addressing scheme can be generalized to other situations and provides a powerful tool for understanding self-similar sets. Note, for example, that the addressing scheme implies that the Cantor set is uncountable.
A major question that we will address later in the book asks, ``What is the dimension of the Cantor set?'' Certainly, it is too small to be considered a one dimensional set; it is just a scattering of points along the unit interval with length zero. It is uncountable, however; perhaps it is too large to be considered as zero dimensional. We will develop a notion of ``fractal dimension'' that quantitatively captures this in-betweeness.