Dimension theory is quite a sophisticated topic (at least for me), it is fully settled in Shafarevich's book on the first 100 pages.
Shafarevich gives two nice applications of the theory. 1) A proof of Tzen's theorem 2) A proof of existence of a line on a cubic surface in $\mathbb P^3$.
Both applications are geometric statements. I would like to learn about some other applications that are easy to state. I am asking this partially because of an introductory course in algebraic geometry that I am teaching. I would like to give some motivation for developing the dimension theory.