Before we begin edit
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 Product edit
Let be an indexed set, and let be a set for each . The Cartesian product of each is
Let and for each . Then
Product Topology edit
Using the Cartesian product, we can now define products of topological spaces.
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.
- 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 :