What is the point of disjoint union?

Question

Disjoint union is seemingly ubiquitous in mathematical literature but it is rarely ever used properly. It is almost always the case that one forms $\bigsqcup_{i\in I} X_i$ and ends ups identifying $x\in X_i$ with $(x,i)\in\bigsqcup_{i\in I} X_i$. (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 $X_i\times\{i\}$ when we don't ever use the notation $(x,i)$?

Answer 1

As Brian said, the formal definition of disjoint union is important because it tells us that such a construction ("assume that the $X_i$ are all pairwise disjoint") is possible. Similarly, it's important to have a formal definition of ordered pairs (usually $(a,b) = \{\{a\}, \{a,b\}\}$). If there was no way to formally construct these things, it would be problematic to use them as mathematical objects! And, beyond telling us that these constructions make sense mathematically, these definitions allow us to prove things – without a formal definition of "ordered pair" it is impossible to prove that $(a,b) = (c,d) \iff a = c \land b = d$.

But the formal definition of these constructions is often separated from our mental models of these objects. Personally, I don't think of ordered pairs as sets of the form $\{\{a\}, \{a,b\}\}$ (even though, for me, this is what they technically are). Instead, I usually think of an ordered pair abstractly, as some kind of data structure that encodes two objects in order, without worrying about exactly how this is defined set-theoretically. This is fine, because I do have a formal definition of "ordered pair" to fall back on if necessary, and my intuition about ordered pairs is very strong (it comes from facts I've proven (long ago) from the formal definition).

I am confident that I will never make an incorrect claim about ordered pairs, and that I will never confuse readers by not using the formal definition of ordered pairs, even though I never deal with the formal definition in practice! Indeed, I think it would be much more confusing to most readers if I did use the formal definition of ordered pairs all the time – the important thing about a particular ordered pair is not that it's a set, but that it contains the data of two mathematical objects, in order.

Likewise, it's important that there is a formal definition of disjoint union. We can use this formal definition to prove things about disjoint unions, and thus build up our intuitions about how they behave. But our mental model of a disjoint union should not always be a collection of tuples; instead we should think of the disjoint union of a bunch of sets in the way you described: take all the sets $X_i$, "make them disjoint", and then stick them together. As such, it's often easier for a reader to digest an argument when we refer to elements of $\coprod_i X_i$ by the same names that they had in the sets $X_i$.

That's not to say that it's never a good idea to write $(x,i)$. For example, if all the sets in the coproduct are equal, it's always necessary to disambiguate the index. That is, I would always write $(x,i)$ for an element of $\coprod_{i \in I} X$.