Topology MOC

Subspace topology

The subspace topology is a natural way of reframing a subspace as a whole space. Let be a topological space, and be any subset with the canonical inclusion . The subspace topology on is the coarsest topology for which the canonical inclusion is continuous.¹ topology

More generally, if is an injective map, then the subspace topology induced by is the coarsest topology for which is continuous. In this case is an embedding.

Further characterisations

Explicit

Let be a topological space and be a subset. A subset is then open relative to iff there exists an open subset (relative to ) such that .² The system of all subsets open relative to is called the subspace topology induced by , and forms a topological space.

Universal property

For every topological space and every map , then is continuous iff is continuous. topology

Proof

First we will prove that the subspace topology as characterised above satisfies the universal property. Let be a topological space and let be a subset endowed with the subspace topology . Let be some topological space, and be a function. If is continuous, then so is the composition of continuous functions. Now suppose is continuous, and let . Then for some . Since is continuous, , thus is continuous. Therefore is continuous iff is continuous.

Now let be a topology on satisfying the universal property. In particular, let and . Then since is continuous so is , wherefore is coarser than Now let with . Since is continuous, so too is . But is the coarsest topology on for which is continuous, therefore .

Properties

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

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Footnotes

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

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