Figure 2.1 Initial steps in the construction of the Cantor set by repeated removal of the middle third of intervals.

of removal operations from a unit-length interval, termed the initiator. This initiator is labeled /0 in Figure 2.1, where it corresponds to the interval [0, 1] (i.e. the set of numbers x such that 0 < x < 1). If one removes from this initiator the segment (1/3, 2/3) (containing the real numbers x such that 1/3 < x < 2/3), the set / results. This set is sometimes termed the generator and consists of the two intervals [0, 1/3] and [2/3, 1]. Removing the middle thirds of these intervals, i.e. applying the generator to each of them, yields /2, which comprises four intervals of length 1/9. At the next stage (/3), there are 23 = 8 intervals of length (1/9)/3 = 3-3. At the nth iteration of this deletion procedure, the set /n consists of 2n intervals, each having a length 3-n, and is included in all the preceding sets /j, /2,..., /„_i in the sequence.

The Cantor set P can be thought of as the limit of the sequence of sets /n when n tends to infinity. Mathematically [9], it is defined as n°=j/„, the intersection of all the sets /n, with n going from one to infinity. It is obviously impossible to draw the set P itself, with its infinitesimal detail, so 'pictures of the Cantor set P' are in fact only illustrations of one of the sets /n. It is apparent from Figure 2.1 that such representations are feasible only for relatively low values of n.

At first glance, it might appear that we have removed so much of the interval [0, 1], during the construction of the Cantor set, that nothing remains. In a sense, this is true. The (Lebesgue) measure m(/n) of the set /n, i.e. practically, the total length of all the 2n intervals included in /n, is given by (2/3)n. Since P = nj;°= j /n and P is included in /n for every n, the measure m(P) is equal to the limit as n ^ to of (2/3)n, which is zero. In other words, the Cantor set has zero length and its topological dimension is zero. Yet it is an uncountable set, containing infinitely many points in any neighborhood of each of its points. Furthermore, one can show that the points

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