According to Engelking (General Topology, p. 76), results regarding disjoint unions are usually very easy to prove and belong to "topological folklore." Most topology textbooks show some rather trivial results regarding disjoint unions, for example:
- disjoint union is a coproduct in the category of topological spaces
- a set is closed in the disjoint union if and only if its intersection with each individual space is closed in that space
- Disjoint union is Hausdorff if and only if each individual space is
I was wondering what were some other interesting (preferably not as trivial as the above) results that use disjoint unions (or coproducts in general).
Any help will be appreciated!