How modern construction of this set. The Cantor

 How large can a set
with zero ‘length’ be?

 

This paper will be a summary of my findings in answering the
questions, “how large can a set with zero ‘length’ be?”. Throughout this paper
I will be explaining facts regarding the Cantor set. The Cantor set is the best
example to answer this question as it is regarded as having length zero. The
Cantor set was discovered in 1874 by Henry John Stephen Smith and it was later
introduced by Gregor Cantor in 1883. The Cantor ternary set is the most common
modern construction of this set.

 

The
Cantor ternary set is constructed by deleting the open middle third, (,) from the interval 0,1, leaving the line segments and . The open middle third of the remaining line segments are
deleted and this process is repeated infinitely. At each iteration of this
process, s of the initial length of the line segment (at that given step)
will be remaining. The total length of the line segments at the nth iteration will therefore be: Ln =
n,
and the number of line segments at this point to be: Nn = 2n.  From this we can also work out that the open
intervals which will be removed by this process at the nth iteration
will be  +   + . . . + .

 

As
the Cantor set is the set of points not removed by the above process it is easy
to work out the total length removed, and from above it is easy to see that at
the nth iteration the length removed is tending towards 1. The total
length removed will therefore be the geometric progression:

 =   +  +   +   + .. =  () = 1.

It
is easy to work out the proportion left is 1 – 1 = 0, suggesting the Cantor set
cannot contain any interval of non-zero length. The sum of the removed
intervals is therefore equal to the length of the original interval.

 

At
each step of the Cantor set the measure of the set is , so we can find that the Cantor set has Lebesgue measure of n at
step n. Since the Cantor set’s construction is an infinite process, we can see
as this measure tends to 0, . Therefore, the whole Cantor set itself has a total measure of 0.

 

There
should, however, be something left as the removal process leaves behind the end
points of the open intervals. Further steps will also not remove these
endpoints, or in fact any other endpoint. The points removed are always the
internal points of the open interval selected to be removed. The Cantor set is
therefore non-empty and contains an uncountable number of elements, however the
endpoints in the set are countable. An example of end points that will not be
removed are  and  , which are the endpoints from
the first step of removal. Within the Cantor set there are more elements other
than the endpoints which are also not removed. A common example of this is   which is contained in the
interval 0. It is easy to tell that there will be infinitely many other
numbers like this example between any two of the closed intervals in the Cantor
set.

 

From
above it is easy to see that the Cantor set contains all the points in the line
segments not deleted by this infinite process in the interval 0,1. As the
construction process is infinite, the Cantor set is regarded to be an infinite
set, i.e. it has an infinite number of elements. The Cantor set contains all
the real numbers in the closed interval 0,1 which have at least one ternary
expansion containing only the digits 0 and 2, this is the result of how the
ternary expansion is written. As it is written in base three, the fraction  will be
equal to the decimal 0.1 (also 0.0222..),  is therefore equal to 0.2
and  equal to 0.01.

 

In
the first step of the construction of the set, we removed all the real numbers
whose ternary decimal representation contain a 1 in the first decimal place,
except for 0.1 itself (this is  and we have found out it is
contained in the Cantor set). Choosing to represent  as 0.222.. this removes all
the ternary decimals that have a 1 in the second decimal place. The third stage
removes those with a 1 in the third decimal place and so on. After all the
numbers have been removed the numbers that are left, i.e. the Cantor set, are those
consisting of ternary decimal representations consisting entirely of 0’s and
2’s.

 

It
is then possible to map every 2 in any number in the Cantor set to a 1, if we
do this it will give the full set of numbers in the interval 0,1 in binary
and therefore mapping the whole of the interval 0,1. This means that there is
a mapping which has its image as the whole of the interval 0,1, meaning that
there is a surjection from the Cantor set to all the real numbers in the
interval 0,1. Since the real numbers are uncountable, the Cantor set must
also be uncountable. The Cantor set must therefore contain as many points as
the set it is made from and it contains no intervals.

 

The
compliment of the Cantor set is made up of the points which are not contained
in the Cantor set, i.e. the points which are removed from the interval 0,1
during the construction of the Cantor set. From above we worked out that the
total length removed was equal to 1, which means the compliment of the Cantor
set must equal 1 as it is defined precisely as that. An example of a number in
the compliment is the number . Like the Cantor set itself, there is an uncountable number of
elements in the compliment. At each step of the cantor set, n, there are n
number of open intervals in the compliment. Between any two endpoints of the
Cantor set it is obvious to point out that there is an entire interval in the
compliment, i.e. the open intervals removed from 0,1 to form the Cantor set.

 

The
Cantor ternary set, talked about above, and in fact the general Cantor set are
examples of fractal sets. A fractal set is a set which is constructed by the
same repeated pattern at every scale. The ternary Cantor set evidentially can
be classed as a fractal set, the pattern demonstrated in the following picture.
The Cantor set split at every step by removing the same fraction of the pattern
at every step and the number of closed intervals doubles as you move to the
next stage of construction. 

  

The
fractal dimension of the Cantor set is  .

 

The
above idea of construction by the ternary method can be generalised to any
other length of removal to form another form of the general Cantor set. The
pattern of forming a generalised Cantor set follows the same construction
patterns as above also. Another interesting fact about the Cantor set is that
there can exist “Cantor dust”. The difference between the two is that Cantor
dust is the multi-dimensional version of a Cantor set. The dust is formed by
taking the finite cartesian product of the Cantor set with itself, this makes
it a Cantor space. The Cantor dust, like the Cantor set, also has a measure of
0.