Topology/Product Spaces

< Topology
Topology
 ← Order Topology Product Spaces Quotient Spaces → 


Contents

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  :

 


Topology
 ← Order Topology Product Spaces Quotient Spaces →