I'm in understanding the intuition behind the notion of proper morphisms and also how people think about it.
https://en.wikipedia.org/wiki/Proper_morphism
A morphism of schemes $f:X \rightarrow Y$ is proper if it is separated, finite type, and universally closed.
I also know it generalizes the notion of a proper map in topological spaces. https://en.wikipedia.org/wiki/Proper_map
But what is not clear to me why separated and universally closed are the right ways to generalize this.
Also, how do people generally think about proper morphisms?