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 1 The 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 2 The 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 .