Topology, A Base for the Topology

A Base for the Topology

Specifying every open set in a topology is often prohibitive. Instead, we usually define a base, an initial collection of open sets, and let union do the rest. The standard topology in the plane appears when the base consists of open disks of all radii, centered at all points. Actually it is sufficient to restrict the disks to rational radii and rational centers. Every open blob in the plane is now a union of these open disks. The standard topology for 3 dimensional space has a base of open balls, and so on.

If a collection of open sets forms the base for a valid topology, it should satisfy the "base criterion", as follows.

If X and Y are two base sets, and X and Y intersect in a point p, then X and Y both contain a base set Z, which contains p.

If the base criterion fails, then X∩Y cannot be a union of base sets, and is not open; yet it must be open, as it is the intersection of two open sets. Thus the base criterion is necessary. We will now show it is sufficient. That is, the base sets generate a valid topology.

First intersect two base sets. If there is a base set Z for every p in X∩Y, then X∩Y is the union of base sets, and is open, as it should be.

Next intersect finitely many base sets. If p is contained in finitely many base sets, let X and Y be the first two, and replace X and Y with Z, a base set in X∩Y that contains p. If the next base set is W, then find a base set in W and Z that contains p. Continue this process until p is contained in a base set that is contained in our finite list of base sets. The finite intersection of base sets is thus a union of base sets, and is open, as it should be.

The union of base sets is open, because that's how we define the topology generated by the base. Obviously the union of open sets is still the union of base sets, and is open. We must show the finite intersection of open sets is open. That is, the finite intersection of arbitrary unions of base sets is covered by base sets.

Let p be a point in this finite intersection. it is in the finite intersection of individual base sets, one base set taken from each union. There may be many ways to do this; we will persue them all in parallel. That way we don't need the axiom of choice. Surround p with a base set Z that is contained in the finite list of base sets. This base set Z covers p, and is in the intersection. Include every Z, for every finite list of base sets that contains p, for every p in the intersection of open sets. The union of all these base sets Z is the finite intersection of our original open sets, hence the finite intersection of open sets is open, and the base generates a valid topology.