This week’s post is by Laura Irvine, who is just about to start her second year reading Mathematics at Murray Edwards College, Cambridge. Thanks very much, Laura!
First of all, what is the Cantor set?
To form the Cantor set, start with the closed interval (this means and are included in the interval) and remove the middle open third of the interval (i.e. remove (, ) where the curved brackets mean the interval is open, so and are not themselves in the interval). You should be left with two disjoint closed intervals: and . I’m going to call this step the first iteration.
Then do the same thing to each of these intervals: remove the middle third of each to get the new intervals as , , , . Then remove the middle third of each of these intervals. Keep repeating this process and the Cantor set is the set of all the points in the interval that are never removed. So the first few steps of the process look like this:
Do you understand how this set is formed? Can you think of some points that are in the Cantor set?
Well, will never be removed: the first closed interval after the iteration would be so will be in the infinite intersection of the first interval of each step. The other interval endpoints are points in the set too. It turns out that there are also points that are in the set that aren’t interval endpoints.
An interesting and, in my opinion, rather surprising property of the Cantor set is that it has measure , despite being an uncountable set! The fact it is uncountable means there is no way of writing all the numbers in the Cantor set in a list.
So, intuitively, this is saying that if all the points in the Cantor set were lined up next to each other, the line would have length and yet there are infinitely many points in the set. How weird is that?!