The area of math that I really like to study is called *descriptive set theory*. In calculus, you often use different types of sets like open sets and closed sets. You also use different types of functions like continuous and differentiable functions. In descriptive set theory, we study more complicated sorts of sets and functions.

You can read much more about descriptive set theory on the wikipedia article here.

Recall that a set is called *open* if it can be written as a union of open intervals. A set is *closed* if its complement is open. Now is where it starts to get complicated.

**Definition**. A set is called $\Sigma_1$ if it is a union of countably many closed sets. A set is called $\Pi_1$ if its complement is $\Sigma_1$. A set is called $\Sigma_2$ if it is a union of countably many $\Pi_1$ sets. A set is called $\Pi_2$ if its complement is $\Sigma_2$. And so on!

Thus, a set is $\Sigma_n$ if it is a union of complements of unions of complements of unions (blah blah) of closed sets! The bigger $n$ is, the more complicated the set! A similar definition exists for functions.

What are the applications? Well, a lot of theorems assert the existence of sets and functions, but never say how complicated these are. Descriptive set theory gives you a way of saying: “Sure, that function exists, but it is way too complex to ever write down explicitly!”

For example, a bizarre theorem of Banach and Tarski says that the unit sphere can be partitioned into 10 sets, which can then be rearranged to form *two* spheres. This certainly sounds counterintuitive, and even a little crazy. Don’t believe me? Read more here! Descriptive set theory has an answer: these 10 subsets are too complicated to ever be described explicitly!