Disjoint union of two sets
Let and be sets. Then their disjoint union is the union
where is an object chosen so that and are disjoint. Normally is identified with in the obvious way. The element is almost never mentioned; it serves only as a “tag” to make the two sets disjoint. If and are already disjoint, then is isomorphic to ; this is the most common situation in practice.
Disjoint union of many sets
Observe that we have a natural isomorphism , and that the images of any pair of these isomorphisms have empty intersection. This is also often called being pairwise disjoint and is a much stronger condition than that the intersection of all the images is empty. As before, if the are already pairwise disjoint, then
For example, as sets, is two copies of the real line. As topological spaces, is again two copies of the real line with a topology whose open sets are pairs of real open sets, one for each copy of . This is the coproduct in the category of topological spaces.
Of course, there are many categories where this usage is unnatural. For example, in the category of pointed sets, the coproduct is the disjoint union with the distinguished points identified. In the category of abelian groups, the coproduct is the direct sum.
Another closely related usage should be mentioned. Occasionally an author will write “…and is a disjoint union…”. What this means is that is isomorphic to , which is to say that and are already disjoint.
|Date of creation||2013-03-22 14:13:24|
|Last modified on||2013-03-22 14:13:24|
|Last modified by||yark (2760)|