## Before we beginEdit

We quickly review the set-theoretic concept of Cartesian product here. This definition might be slightly more generalized than what you're used to.

## Cartesian ProductEdit

### DefinitionEdit

Let be an indexed set, and let be a set for each . The **Cartesian product** of each is

## ExampleEdit

Let and for each . Then

## Product TopologyEdit

Using the Cartesian product, we can now define products of topological spaces.

### DefinitionEdit

Let be a topological space. The **product topology** of is the topology with base elements of the form , where for all but a finite number of and each is open.

## ExamplesEdit

- Let and with the usual topology. Then the basic open sets of have the form :

- Let and (The Sorgenfrey topology). Then the basic open sets of are of the form :