The Cantor Set

The Cantor set is a relatively simple set, involving only the real numbers between 0 and 1 inclusive

It is made by deleting the open middle third of the line section (as in leaving the 2 points 1/3 and 2/3), and then deleting the middle third of each of the line segments left over ad infinitum.

The cantor set itself is comprised of all the points in the interval that have not been deleted at any step during this process. Although the process goes on infinitely, which may leave you to believe that no points will be left over in the end. But there will be some points left over.

Why?

Well, remember that the open middle third is deleted so the points 1/3 and 2/3 remain. Those points will never be deleted, as they will always be on the end of any future lines so they will never be contained in any open middle third, in fact nor will the endpoint of any line segment created. So there will always be points in the Cantor set.

However, if we consider the parts we remove, frst removing 1/3, then 2/9, etc and sum that to infinity:

We see that the total length removed is 1, but we only started with 1! So according to this, there is nothing left...
But we just showed there must be some points remaining...
So any intervals remaining must have a length of 0.
It's curious facts like this that I was referring to earlier.