Topology MOC

Coproduct topology

The coproduct topology or sum topology is the canonical way of defining a topology on the Disjoint union of spaces. Let {(𝑋𝛼,T𝛼)}𝛼∈𝐴 be an arbitrary collection of topological spaces with disjoint union

𝑋=∐𝛼∈𝐴𝑋𝛼

and 𝜄𝛼 :𝑋𝛼 ↣𝑋 as canonical inclusions. The coproduct topology on 𝑋 is the finest topology on 𝑋 for which all inclusions 𝜄𝛼 are continuous.¹ topology

T𝑋={𝑈⊆𝑋:𝜄−1𝛼𝑈∈T𝛼∀𝛼∈𝐴}

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.

T𝑋={⋃𝛼∈𝐴𝜄𝛼𝑈𝛼:𝑈𝛼∈T𝛼}={∐𝛼∈𝐴𝑈𝛼:𝑈𝛼∈T𝛼}

Bases for the coproduct topology

The images of open sets form a topological basis:

B𝑋={𝜄𝛼𝑈𝛼:𝑈𝛼∈T𝛼:𝛼∈𝐴}

It follows that if B𝛼 is a basis of 𝑋𝛼 for each 𝛼 ∈𝐴, the following is also a basis of 𝑋:

B′𝑋={𝜄𝛼𝑈𝛼:𝑈𝛼∈B𝛼:𝛼∈𝐴}

Universal property for the coproduct topology

For every topological space (𝑍,T𝑍) and function 𝑓 :𝑋 →𝑍, then 𝑓 is continuous iff 𝑓𝜄𝛼 :𝑋𝛼 →𝑍 is continuous for all 𝛼 ∈𝐴. topology

Proof

First we will prove that the coproduct topology as characterised above satisfies the universal property. Let {(𝑋𝛼,T𝛼)}𝛼∈𝐴 be topological spaces and let 𝑋 =∐𝛼∈𝐴 be their disjoint union endowed with the coproduct topology and with canonical inclusions 𝜄𝛼 :𝑋𝛼 ↣𝑋. Let (𝑍,T𝑍) be some topological space, and let 𝑓 :𝑋 →𝑍 be a function. If 𝑓 is continuous, then so are the compositions 𝑓𝜄𝛼 of continuous functions for all 𝛼 ∈𝐴. Now suppose 𝑓𝜄𝛼 :𝑋𝛼 →𝑍 is continuous for all 𝛼 ∈𝐴, and let 𝑈 ∈T𝑍. Then (𝑓𝜄𝛼)−1𝑈 =𝜄−1𝛼𝑓−1𝑈 ∈T𝛼 for all 𝛼 ∈𝐴, whence 𝑓−1𝑈 ∈T𝑋. Thus 𝑓 is continuous. Therefore 𝑓 is continuous iff 𝑓𝜄𝛼 are continuous for all 𝛼 ∈𝐴.

Now let T′ be a topology on 𝑋 satisfying the same universal property. In particular, let (𝑍,T𝑍) =(𝑋,T𝑋) and 𝑓 =id𝑋 :(𝑋,T′) →(𝑋,T𝑋). Then since id𝑋⁡𝜄𝛼 is continuous for all 𝛼 ∈𝐴, so is id𝑋 :(𝑋,T′) →(𝑋,T𝑋), wherefore T′ is finer than T𝑋 Now let (𝑍,T𝑍) =(𝑋,T𝑋) and 𝑓 =id𝑋 :(𝑋,T′) →(𝑋,T′). Since id𝑋 is continuous, so too is id𝑋⁡𝜄𝛼 =𝜄𝛼 for all 𝛼 ∈𝐴. But T𝑋 is the finest topology for which all 𝜄𝛼 are continuous, so T′ =T𝑋

Spaces constructed as coproducts

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

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Footnotes

1. 2020, Topology: A categorical approach, pp. 32–33 ↩