The Cantor Set
Consider the unit interval in the real line, , and remove the “middle third open subinterval” . We have the closed set , which is the union of two closed intervals. In a second step, remove the “middle third open subintervals” of each of the previous two intervals in , to obtain the closed set .
The image below illustrates this procedure with the three sets constructed so far.
We iterate this procedure, thus constructing on each step a new closed set by removal of “middle third open subintervals” from the previous set . Equivalently, the operation can be described as: “take the union of the previous set, with a shift of it by two units, and scale them down by one-third.” We can write then for each Notice that
This process is permitted to continue indefinitely. At the limit, we have removed from the unit interval a countable number of open subintervals; therefore, the resulting set must be closed. This is what we call the “Cantor set.”
Structure of the Cantor set
Notice that the Cantor set is not empty: the numbers , , , , , , and all belong in . As a matter of fact, if , then either or (or both!) which indicates that there are infinitely many elements in the Cantor set. We will use this fact below.
Let us measure the Cantor set: The unit interval measures one unit. In the first step, we have removed a subinterval of size ; hence, measures . In the second step, we have removed from two subintervals of size ; hence measures units. In the th step, we remove from as many as subintervals of size ; therefore, the size of is units. At the limit, the Cantor set has therefore size
How is that possible?
By definition, if and only if for all This means, in particular, that we can find such that for any , there will be an –tuple with satisfying
Equivalently, for each and any there is some and an -tuple with such that It should be no trouble to prove that this property defines every element of the Cantor set.
For example, consider the sequence with and for all . The sequence of partial sums is convergent, with
Indeed, notice that
We have them proven that is in the Cantor set, although it is not the endpoint of any of the removed subintervals in the original construction!
This is an alternative way to describe all numbers in the Cantor set:
if and only there exists a sequence with for all , such that the partial sum sequence converges to
We will use this description to construct an injective map thus proving that the cardinality of the Cantor set is the same as that of the unit interval (and therefore, the Cantor set is uncountable!)
Let us use the density of dyadic fractions: for each and , find a dyadic number with such that We write in its (unique) base-two expression as with values for all It is then
A similar reasoning as above shows that any number in the unit interval can be realized as the limit of a partial sum of a sequence for a sequence satisfying for all We construct the function as follows: