Constructions on Topological Spaces

Subspaces

Subspace Topology

If is a topological space and , then the subspace topology on is We call a topological subspace of .

e.g. If we consider as a subspace of then the open sets in consist of all sets where is an open subset of . In particular, is open in the subspace for every .

Lemma

Suppose that is a metric space with corresponding topology . If then subspace topology on corresponds to the topology on that arises from the metric space .

Proof

Proposition

Closed subset in closed subspace is closed in the original topological space.

Proof Let be a closed subspace of , and is closed in . Then for some open set in . It follows that . Thus, Since is closed in , is open. Therefore is open, and is closed in .

Universal Property of the Subspace Topology

Let be a topological space and a subspace (with the subspace topology). Then

1. The inclusion map is continuous;
2. For any topological space , a map is continuous iff the composite is continuous;
3. The subspace topology is the unique topology on with this property.

Proof The continuity of the inclusion map follows from 2. For 2, suppose is continuous, then for any open set , we have , which is open in . The converse is similar. To show such topology is unique, suppose is another topology on with the same property. Then the identity map is continuous. Thus for any open set , we have . Similarly, we can show that . Therefore, .

Corollary

Suppose is continuous. Then

• The restriction is continuous for any subspace ;
• If is a subspace of that contains the image , then is continuous;
• If is a subspace of , then is continuous.
  

Proof This is obvious from the above theorem.

Product Spaces

Product Topology

Suppose that and are two topological spaces. Then the product topology on is the topology with basis
We call the product topology on .

Remark

Note that may not be a topology on . For example.
Moreover, the product topology is independent of choice of the bases. This is because if and are bases for and , respectively, then the topology generated by is the same as the topology generated by . (This is a direct result of proposition)

the topology generated by A0 ⇥ B0 is the same as the topology generated by

A ⇥ B

Thrm Let and be topological spaces with respective bases and . Then

Quotient Topology

Quotient Topology

Suppose is a topological space and is a set. is surjective, then the quotient toplogy is a topology on by declaring that is open iff is open in .

Universal Property of Quotient Topology

The quotient topology has the following universal property (in ):