Equality vs Equivalence in Mathematics

In algebraic geometry, what we're thinking about is not topological spaces, but sort of geometrical spaces. And the difference between geometry and topology is really what we count as the same. So there is a sense in which we can count a coffee cup and a doughnut as being the same. Of course, there's also another sense in which they're not the same. But we get to make those choices in math. I think there's definitely not enough emphasis in mainstream math education on our ability to make choices. That choice is usually not given to students of math.