We are taught since when we are young that schemes are cool because they take into account "nilpotents". This means also that we can distinguish between schemes which have the same underlying topological space but for example in the sense that they differ by some "thickening".

I would like to get some intuition and motivation about the notion of universal homeomorphism: by definition a morphism of schemes $f\colon X\to Y$ is a $\mathit{universal\,\, homeomorphism}$ if for every morphism of schemes $Z\to Y$ the pull-back morphism induces a homeomorphism $\lvert Z\times_Y X \rvert \to \lvert Z\rvert$.

I haven't been able to find some discussion about that, and in the stacks project (28.43) they almost apologize for including that section because a universal homeo "is really just an integral, universally injective and surjective morphism". I also have the feeling that they are taken into account especially when one works in positive characteristic.

This might be the question: do we need universal homeomorphisms? do we really need universal homeomorphisms when working over $\mathbb{C}$?

Thank you!