Cantor set

The Cantor setMathworldPlanetmath C is the canonical example of an uncountable set of measure zeroMathworldPlanetmath. We construct C as follows.

Begin with the unit interval C0=[0,1], and remove the open segment R1:=(13,23) from the middle. We define C1 as the two remaining pieces

C1:=C0R1=[0,13][23,1] (1)

Now repeat the process on each remaining segment, removing the open set

R2:=(19,29)(79,89) (2)

to form the four-piece set

C2:=C1R2=[0,19][29,13][23,79][89,1] (3)

Continue the process, forming C3,C4, Note that Ck has 2k pieces.

Also note that at each step, the endpoints of each closed segment will stay in the set forever—e.g., the point 23 isn’t touched as we remove sets.

The Cantor set is defined as

C:=k=1Ck=C0n=1Rn (4)

Cardinality of the Cantor set

To establish cardinality, we want a bijection between the Cantor set and some set whose cardinality we know .

Start at C1, which has two pieces. Mark the left-hand segment “0” and the right-hand segment “1”. Then continue to C2, and consider only the leftmost pair. Again, mark the segments “0” and “1”, and do the same for the rightmost pair.

Keep doing this all the way down the Ck, starting at the left side and marking the segments 0, 1, 0, 1, 0, 1 as you encounter them, until you’ve labeled the entire Cantor set.

Now, pick a path through the tree starting at C0 and going left-left-right-left…and so on. Mark a decimal point for C0, and record the zeros and ones as you proceed. Each path has a unique number based on your decision at each step. Every point in the Cantor set will have a unique address dependent solely on the pattern of lefts and rights, 0’s and 1’s, required to reach it. The Cantor set therefore has the same cardinality as the set of sequences of 0’s and 1’s, which is 20, the cardinality of the continuumMathworldPlanetmath.

The Cantor set and ternary expansions

Return, for a moment, to the earlier observation that numbers such as 13, 29, the endpoints of deleted intervals, are themselves never deleted. In particluar, consider the first deleted interval: the ternary expansions of its constituent numbers are precisely those that begin 0.1, and proceed thence with at least one non-zero “ternary” digit further along. Note also that the point 13, with ternary expansion 0.1, may also be written 0.02˙ (or 0.02¯), which has no ternary digit 1. Similar descriptions apply to further deleted intervals. The result is that the Cantor set is precisely those numbers in the set [0,1] that have a ternary expansion contains no digits 1.

Measure of the Cantor set

Let μ be Lebesgue measureMathworldPlanetmath. The measureMathworldPlanetmath of the sets Rk that we remove during the construction of the Cantor set are

μ(R1) =23-13=13 (5)
μ(R2) =(29-19)+(89-79)=29 (6)
μ(Rk) =n=1k2n-13n (8)

Note that the R’s are disjoint, which will allow us to sum their measures without worry. In the limit k, this gives us

μ(n=1Rn)=n=12n-13n=1. (9)

But we have μ(C0)=1 as well, so this means

μ(C)=μ(C0n=1Rn)=μ(C0)-n=112n=1-1=0. (10)

Thus we have seen that the measure of C is zero (though see below for more on this topic). How many points are there in C? Lots, as we shall see.

So we have a set of measure zero (very tiny) with uncountably many points (very big). This non-intuitive result is what makes Cantor sets so interesting.

Cantor sets with positive measure

Clearly, Cantor sets can be constructed for all sorts of “removals”—we can remove middle halves, or thirds, or any amount 1r,r>1 we like. All of these Cantor sets have measure zero, since at each step n we end up with

Ln=(1-1r)n (11)

of what we started with, and limnLn=0 for any r>1.

However, it is possible to construct Cantor sets with positive measure as well; the key is to remove less and less as we proceed. These Cantor sets have the same “shape” (topology) as the Cantor set we first constructed, and the same cardinality, but a different “size.”

Again, start with the unit interval for C0, and choose a number 0<p<1. Let

R1:=(2-p4,2+p4) (12)

which has length (measure) p2. Again, define C1:=C0R1. Now define

R2:=(2-p16,2+p16)(14-p16,14+p16) (13)

which has measure p4. Continue as before, such that each Rk has measure p2k; note again that all the Rk are disjoint. The resulting Cantor set has measure

μ(C0n=1Rn)=1-n=1μ(Rn)=1-n=1p 2-n=1-p>0.

Thus we have a whole family of Cantor sets of positive measure to accompany their vanishing brethren.

Title Cantor set
Canonical name CantorSet
Date of creation 2013-03-22 12:21:37
Last modified on 2013-03-22 12:21:37
Owner yark (2760)
Last modified by yark (2760)
Numerical id 34
Author yark (2760)
Entry type Definition
Classification msc 28A80
Related topic CountableMathworldPlanetmath
Related topic Measure
Related topic SingularFunction
Related topic CantorFunction
Related topic MengerSponge
Related topic RepresentationTheoremForCompactMetricSpaces