What is "disjoint union"

n. 1 (context mathematics English) A (l/en: union) of (l/en set sets) forced to be disjoint by attaching information referring to the original sets to their elements. 2 (context mathematics English) A (l/en: union) of sets which are already disjoint.

In set theory, the disjoint union (or discriminated union) of a family of sets is a modified union operation that indexes the elements according to which set they originated in. Or slightly different from this, the disjoint union of a family of subsets is the usual union of the subsets which are pairwise disjoint – disjoint sets means they have no element in common.

Note that these two concepts are different but strongly related. Moreover, it seems that they are essentially identical to each other in category theory. That is, both are realizations of the coproduct of category of sets.

In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, two or more spaces may be considered together, each looking as it would alone.

The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction.