I just read about quotient spaces, and I'm quite fascinated to have learned that those "images" that I would see online which involve "visualizing" some two-dimensional shape like a torus as a unit square with some funky business going on at the border are actually quite simple to formalize using basic topology and equivalence classes.

The idea is that if we define an equivalence class on a topological space $(X, \tau)$, then there is a unique topology on $X/{\sim}$ such that a subset of equivalence classes is open if and only if its union (which is a subset of $X$) is open in $X$. So when we take a unit square and write "arrows" at the border, what we're really doing is describing visually what the equivalence relation is!

I'm too lazy to put a picture here, I just wanted to share my joy.