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