# general topology – What is the point of disjoint union?

Disjoint union is seemingly ubiquitous in mathematical literature but it is rarely ever used properly. It is almost always the case that one forms $$bigsqcup_{iin I} X_i$$ and ends ups identifying $$xin X_i$$ with $$(x,i)inbigsqcup_{iin I} X_i$$. (Open any topology textbook and you will see this abuse.) Why not just assume that the $$X_i$$ are all pairwise disjoint and instead use ordinary union? Why bother with $$X_itimes{i}$$ when we don’t ever use the notation $$(x,i)$$?