Time for some basics of point-set topology. Let be a nonempty set. A collection of subsets of is called a **topology** on , and the pair a **topological space** if

- ,
- ,
- .

The members of are called **open sets**. The collection is called the **trivial topology**, and the power set of ,

is the **discrete topology**. If and are topologies on with , then is said to be **coarser** than , and is said to be **finer** than .

Theorem 1The intersection of topologies is a topology.

*Proof:* Let be topologies on , for . Put . Since for each , . If , then for each , so for each , and hence . If , for then , so for each and hence . It follows that is a topology.

A simple corollary of this statement is that every set has a unique **generated topology**, namely the intersection of all topologies on containing :

We say that a collection of subsets of is a **basis** for a topology on if

- For each , there exists such that .
- If and with , then there exists such that .

The **topology generated by ** is defined as

Let’s verify that this actually defines a topology!

Theorem 2The topology generated by a basis is indeed a topology.

*Proof:* The statement holds vacuously. For any , there exists such that , so . If then if , there exist such that and . Hence there exists such that , so that . If , , let . If , then for some , so there exists with . It follows that .