Coproduct topology

The coproduct topology or sum topology is the canonical way of defining a topology on the Disjoint union of spaces. Let be an arbitrary collection of topological spaces with disjoint union

and as canonical inclusions. The coproduct topology on is the finest topology on for which all inclusions are continuous.¹ #m/def/topology

Further characterisations

Explicit

The open sets in the coproduct topology correspond exactly to the unions of images of open sets in the constituent topologies, i.e.

Bases for the coproduct topology

The images of open sets form a topological basis:

It follows that if is a basis of for each , the following is also a basis of :

Universal property for the coproduct topology

For every topological space and function , then is continuous iff is continuous for all . #m/thm/topology

Proof

First we will prove that the coproduct topology as characterised above satisfies the universal property. Let be topological spaces and let be their disjoint union endowed with the coproduct topology and with canonical inclusions . Let be some topological space, and let be a function. If is continuous, then so are the compositions of continuous functions for all . Now suppose is continuous for all , and let . Then for all , whence . Thus is continuous. Therefore is continuous iff are continuous for all .

Now let be a topology on satisfying the same universal property. In particular, let and . Then since is continuous for all , so is , wherefore is finer than Now let and . Since is continuous, so too is for all . But is the finest topology for which all are continuous, so

Spaces constructed as coproducts

The Discrete topology is the coproduct of all its points viewed as singletons.

#state/tidy | #lang/en | #SemBr

2020, Topology: A categorical approach, pp. 32–33 ↩