The Axiom of Choice states that we can always form a set by choosing one element from

each set in a collection of non-empty sets. Since its introduction in 1904, this seemingly

simple statement has been somewhat controversial because it is magically powerful in

mathematics in general and topology in particular. In the talk, we will look at what

topology is and we will introduce some important topological properties such as

compactness. Next, we will state one major equivalent to the Axiom of Choice in

topology --- Tychonoff’s Theorem --- which asserts that the product of any collection of

compact topological spaces is compact. Our main goal is to give a proof of the Axiom of

Choice from Tychonoff’s Theorem. This proof was first introduced by Kelley in 1950;

however, it was slightly flawed. We will go over Kelley’s initial proof and we will give the

known correction to his proof. Finally, we introduce the Boolean Prime Ideal Theorem (a

weaker version of the Axiom of Choice), which is equivalent to Tychonoff’s Theorem for

Hausdorff spaces.