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!