I'm currently reading through an introduction to topology book in which the first chapter is an overview of set theory. In this chapter, the Cartesian Product of two sets:
$$A \times B $$
is discussed, which seems relatively straightforward. The author then goes on to say that one can generalize the Cartesian Product to any number of sets with the Direct Product: $$\prod_{i=1}^n A_i = A_1 \times A_2 \hspace{1mm} \times \hspace{1mm} ... \hspace{1mm} \times \hspace{1mm} A_n $$
Which, again, at face value seems relatively straightforward. However, I feel as though there's much more to the direct product than it just being an extension of the Cartesian Product. This being said, what is the difference between the two? Is it true that the Cartesian Product is really the special case of the Direct Product when $n=2$? I would really appreciate it if anyone can help me out! Thanks!
Your author's use of the term "direct product" here is unusual. Most people would also call their $n$-fold product the "Cartesian product". There is no difference between the binary Cartesian product and this general $n$-fold "direct product" in the case $n=2$.
The distinction between "direct product" and "Cartesian product" instead normally refers to what kind of structures you are talking about. "Cartesian product" usually means that you are just talking about sets with no additional structure. "Direct product" usually means you have some sort of algebraic structure on each set and are considering the Cartesian product to have the same algebraic structure defined coordinatewise. For instance, if $(A,\cdot_A)$ and $(B,\cdot_B)$ are groups, their direct product is the group whose underlying set is the Cartesian product $A\times B$ with the group operation defined by $(a,b)\cdot (c,d)=(a\cdot_A c,b\cdot_B d)$.