A disjoint union, or coproduct, is a space such that given two spaces , a continuous map from has the same data as a map from each of and . Compare this to a product of topological spaces which, in the standard product topology, where a continuous map some is the same data as a continuous map to each of .
Disjoint union
Given a collection of sets , their disjoint union, or coproduct, is
For a single set , while the usual union of with itself is simply the original set , while involves two distinct copies of .
Further, there exist natural injective inclusion maps given by .
Note that if is discrete and is any space, then is a homeomorphism.