The Cantor set, named after the German mathematician Georg Cantor (1845-1918), i

Question

The Cantor set, named after the German mathematician Georg Cantor (1845-1918), is constructed as follows. We start with the closed interval [0, 1] and remove the open interval (1/3, 2/3). That leaves the two intervals [0, 1/3] and [2/3, l] and we remove the open middle third of each. Four intervals remain and again we remove the open middle third of each of them. We continue this procedure indefinitely, at each step removing the open middle third of every interval that remains from the preceding step. The Cantor set consists of the numbers that remain in [0, 1] after all those intervals have been removed. Show that the total length of all the intervals that are removed is 1. Despite that, the Cantor set contains infi­nitely many numbers. Give examples of some numbers in the Cantor set.

Explanation / Answer

Thus first we remove the open interval (1/3, 2/3). This leaves a union of two closed intervals: [0, 1/3] [2/3, 1]. I'll call the removed interval d1,1. Next we split each of the remaining intervals [0, 1/3] and [2/3, 1] into three smaller ones and remove the middle part d2, 1= (1/9, 2/9) from the first and d2,2 = (7/9, 8/9) from the second, respectively. We are left with the union of four intervals: [0, 1/9][2/9, 1/3][2/3, 7/9][8/9, 1].

Observe that we started with the interval [0, 1] of length 1. After removing d1,1, the total length of the remaining intervals became 2/3. d2,1 and d2,2 each contributed 1/3 to the total. So after their removal, the four remaining intervals had the total length of (2/3)2. Next we remove 4 middle intervals d3,1, ..., d3,4 leaving 8 smaller (closed) intervals with the total length of (2/3)3. The process never stops. In general, on the step number p we remove 2p-1 intervals dp,1, ..., dp,2p-1. The total length of the remaining intervals is (2/3)p. Obviously, as p grows, the length (2/3)p tends to 0. However, this does not mean that C0 is empty. Moreover, the set is not even countable. The most convenient way to see this is by using ternary representation of the decimals from [0, 1]. In the ternary system the only digits allowed are 0, 1, and 2.

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