Topology MOC

Subspace topology

The subspace topology is a natural way of reframing a subspace as a whole space. Let (𝑋,T𝑋) be a topological space, and 𝑌 ⊆𝑋 be any subset with the canonical inclusion 𝑖 :𝑌 ↣𝑋. The subspace topology on 𝑌 is the coarsest topology for which the canonical inclusion is continuous.¹ topology

T𝑌={𝑖−1𝑈:𝑈∈T𝑋}

More generally, if 𝑓 :𝑆 ↣𝑋 is an injective map, then the subspace topology induced by 𝑓 is the coarsest topology for which 𝑠 is continuous. In this case 𝑓 is an embedding.

Further characterisations

Explicit

Let (𝑋,T) be a topological space and 𝑌 ⊆𝑋 be a subset. A subset 𝑉 ⊆𝑌 is then open relative to 𝑌 iff there exists an open subset 𝑈 (relative to 𝑋) such that 𝑉 =𝑈 ∩𝑌.² The system T𝑌 of all subsets open relative to 𝑌 is called the subspace topology induced by 𝑋, and (𝑌,T𝑋) forms a topological space.

Universal property

For every topological space (𝑍,T𝑋) and every map 𝑓 :𝑍 →𝑌, then 𝑓 is continuous iff 𝑖𝑓 :𝑍 →𝑋 is continuous. topology

Proof

First we will prove that the subspace topology as characterised above satisfies the universal property. Let (𝑋,T𝑋) be a topological space and let 𝑌 ⊆𝑋 be a subset endowed with the subspace topology T𝑌. Let (𝑍,T𝑍) be some topological space, and 𝑓 :𝑍 →𝑌 be a function. If 𝑓 is continuous, then so is the composition 𝑖𝑓 of continuous functions. Now suppose 𝑖𝑓 :𝑍 →𝑋 is continuous, and let 𝑈 ∈T𝑌. Then 𝑈 =𝑖−1𝑉 for some 𝑉 ∈T𝑋. Since 𝑖𝑓 is continuous, 𝑓−1𝑈 =(𝑖𝑓)−1𝑉 ∈T𝑍, thus 𝑓 is continuous. Therefore 𝑓 is continuous iff 𝑖𝑓 is continuous.

Now let T′ be a topology on 𝑌 satisfying the universal property. In particular, let (𝑍,T𝑍) =(𝑌,T𝑌) and 𝑓 =id𝑌 :𝑦 ↦𝑦. Then since 𝑖id𝑌 =𝑖 is continuous so is id𝑌, wherefore T′ is coarser than T𝑌 Now let (𝑍,T𝑍) =(𝑌,T′) with 𝑓 =id𝑌. Since id𝑌 is continuous, so too is 𝑖id𝑌 =𝑖. But T𝑌 is the coarsest topology on 𝑌 for which 𝑖 is continuous, therefore T′ =T𝑌.

Properties

• The subspace is closed iff 𝑖 is a closed map
• The subspace is open iff 𝑖 is an open map

---

tidy| en | SemBr

Footnotes

1. 2020, Topology: A categorical approach, p. 25–26 ↩

2. 2010, Algebraische Topologie, p. 9 (Definition 1.2) ↩