This is a post regarding the basic properties of the Cantor set.
We start with the definition: The Cantor set is obtained by first constructing a sequence of closed sets and then taking the intersection of the sets in this sequence. The sequence is constructed as follows:
1. Start with and remove the middle third, i.e. the interval to obtain the set . So contains disjoint closed intervals each of length .
2. Next remove the middle third of each of these two intervals leaving consisting of disjoint closed intervals each of length .
3. Assuming has been constructed and consists of disjoint closed intervals each of length , remove the middle thirds of all these intervals to obtain consisting of disjoint closed intervals each of length .
By induction we have our sequence .
Now, we define the Cantor set as .
We now describe the properties of the Cantor set:
1. In the usual topology on the Cantor set is closed (being an intersection of closed sets). Moreover since it is bounded so by the Heine Borel theorem it is compact. What’s extraordinary is that every compact metric space is a continuous image of the Cantor set! The proof may be found here.
2. The Cantor set is uncountable. We prove this by considering the ternary expansion of all numbers in . All such numbers may have the form with (including ). We claim that iff has a ternary expansion of the form with .
To prove this consider the construction of through the ternary lens: In ternary the construction of involves removing all numbers in from . This shows that in every number has a ternary expansion of the form with and . Conversely every number of this form is in . Likewise the construction of involves removing all numbers in and in from . This shows that in every number has a ternary expansion of the form with and . Again conversely every number of this form is in . Continuing in this way we can conclude that iff with and . Hence by definition of our claim is established.
Now assume that is countable and has been enumerated in the list described below:
Here each .
Now let where if and otherwise. Then despite being of the requisite ternary form. Hence is uncountable.
3. The Cantor set has Lebesgue measure zero. We define a set to be of Lebesgue measure zero if a sequence of intervals such that and . Here if the end points of are and (with then . (The proof that this definition is consistent with the Lebesgue measure may be found in this book.) Now given choose so that . Since and the total length of is , so it is clear that is a set of Lebesgue measure zero. Together with point 2, we see that this establishes the existence of uncountable sets of zero measure!